Plans for in-person meetings
Project to Create Statistics Content Modules | |
---|---|
Project | StatisticsContent |
Design | Objectives | Learning Design |
Resources | Reading in statistics | By type | By topic |
Content | Learn by doing |
Intro Stats course | Syllabus | Online course schedule | Schedule for in-person meetings | Plans for in-person meetings |
GSE Stat Methods II | Review notes | Topic resources |
The Introductory Statistics course includes weekly in-person meetings, to extend the concepts learned via group activities, projects, and discussion of statistics in current events. The detailed plans for each week's session are as follows:
Contents
- 1 Week 39 (June 17)
- 2 Week 38 (June 10)
- 3 Week 37 (Jun 3)
- 4 Week 36 (May 27)
- 5 Week 35 (May 20)
- 6 Week 34 (May 13)
- 7 Week 33 (May 6)
- 8 Week 32 (April 29)
- 9 Week 31 (April 15)
- 10 Week 30 (Apr 8)
- 11 Week 29 (Apr 1)
- 12 Week 28 (Mar 25)
- 13 Week 27 (Mar 18)
- 13.1 Statistics in the big outside world
- 13.2 Review of concepts related to hypothesis testing (pop proportion)
- 13.3 Run hypothesis testing demo
- 13.4 Discuss student-created hypothesis testing scenarios
- 13.5 Discuss possible end-of-course project(s)
- 13.6 Jeopardy -- review of first semester...for fun
- 14 Week 26 (Mar 11)
- 15 Week 25 (Mar 4)
- 16 Week 24 (Feb 25)
- 17 Week 23 (Feb 18)
- 18 Week 22 (Feb 11)
- 19 Week 21 (Feb 4)
- 20 Week 20 (Jan 28)
- 21 Week 19 (Jan 21)
- 22 Week 18 (Jan 14)
- 23 Week 17 (Jan 7)
- 24 Week 16 (Dec 17)
- 25 Week 15 (Dec 10)
- 26 Week 14 (Dec 3)
- 27 Week 12 (Nov 19)
- 28 Week 11 (Nov 12)
- 29 Week 10 (Nov 5)
- 30 Week 9 (Oct 29)
- 31 Week 8 (Oct 22)
- 32 Week 7
- 33 Week 6
- 34 Week 5
- 35 Week 4
- 36 Week 3
- 37 Week 2
- 38 Week 1
- 39 Other Ideas
- 40 References
Week 39 (June 17)
Statistics in the big outside world
As students assemble, discuss instances of statistics and probability used in current media.
Student presentations
Each student presents on their research study for 5-10 minutes.
TED Talk on statistics
If time, show the Ted Talk Lies, Damned Lies, and Statistics (about TedTalks) by Sebastian Wernicke. Length=6 min.
Pre-talk:
- Remind students about data mining as a method for using statistics to make sense of existing data.
- This talk describes results of analyses on a particularly rich data set.
- Ask students to critically evaluate the speakers analysis results and conclusions.
Post-talk:
- Ask students their thoughts on the speaker's analysis results and conclusions. (students should comment that the speaker is jumping from correlation to causation over and over again...that the whole premise sits on this faulty reasoning. But the talk is clearly tongue-in-cheek....so maybe that's the point.)
Week 38 (June 10)
No meeting. Students to complete data collection for their statistics research project.
Week 37 (Jun 3)
Statistics in the big outside world
As students assemble, discuss instances of statistics and probability used in current media.
Concepts missed in the quiz
- What conditions must be met to use the chi-square test? (1. Sample is random; 2. Conservatively, all expected counts are above 5).
- If we fail to reject Ho, can we conclude then that we accept it? (NO! We were building a case for the alternative. When we fail to reject in favor of the alternative, we are NOT saying that we have evidence in favor of Ho.)
End-of-course projects
- Display "Final Student Project -- Step 2: Procedure"
- Have each student present/discuss the procedure for his/her proposed research question.
- Discuss as needed to help student work out the details.
- Discuss next steps:
- collect data (do we need to meet to discuss this? June 9 *Wed*?)
- analyze data and write report (for June 17 session) - do you want some guidance on how to structure the report?
- brief presentation of research results and interpretation (June 17 session)
Week 36 (May 27)
Statistics in the big outside world
As students assemble, discuss instances of statistics and probability used in current media.
End-of-course projects
- Display outline step in "Final Student Project"
- Have each student present/discuss the outline of his/her proposed research question or topic area.
- Discuss as needed to help student work out the details in the plan
- Discuss next steps:
- create procedure (for June 3 session),
- collect data (for June 9 *Wed* session)
- analyze data and write report (for June 17 session)
- brief presentation of research results and interpretation (June 17 session)
Week 35 (May 20)
Statistics in the big outside world
As students assemble, discuss instances of statistics and probability used in current media.
End-of-course projects
- Display outline step in "Statistics Project Directions" (obtained from an ap stats class)
- Have each student present/discuss his/her proposed research question or topic area.
- If the student has a defined research question, have the student present the broad specs for the study (as shown on the outline).
- Discuss as needed to help student narrow focus to a particular question and to broadly specify the study.
- Offer additional work time in groups if students feel that would be useful.
Review of course content
- One at a time, present free-response questions 1-4 from 2009 AP Statistics test.
- Students can collaborate on calculation/creation of solution.
- Discuss solutions as a group.
Week 34 (May 13)
Statistics in the big outside world
As students assemble, discuss instances of statistics and probability used in current media.
Discuss/review concepts in inference for relationships
Each student was asked to develop 2-3 questions based on an assigned topic in the inference for relationships module. Have each student present their questions to the other students for response/discussion.
Review any questions/issues submitted in the "My Response" section of module 12.
Inferences for M&M data
Display the M&M data collected in the first sessions.
- What hypotheses might be interesting to investigate?
- What tests would you use?
- Is the data sufficient to run the tests?
Form two groups, and ask each group to develop and analyze one M&M question, using the...
4-step hypothesis testing process:
- State the appropriate null and alternative hypotheses, Ho and Ha.
- Determine an appropriate test statistic.
- Data collection and exploratory analyses
- Obtain a random sample and collect relevant data.
- Conduct exploratory data analyses.
- Check whether the data meet the conditions under which the test can be used.
- If the conditions are met, summarize the data by a test statistic.
- Find the p-value of the test.
- Based on the p-value, decide whether or not the results are significant and draw your conclusions in context.
End-of-course projects
- Have each student present/discuss possible topic(s).
- Share "Statistics Project Directions" (obtained from an ap stats class) to provide structure for experimental design process.
- Break into two groups for more specific sharing and discussion of potential projects.
Week 33 (May 6)
Statistics in the big outside world
As students assemble, discuss instances of statistics and probability used in current media.
Discuss concepts in inference for relationships: chi-square
2 students were asked to develop 2-3 questions based on an assigned set of screens. Have each student present their questions to the other students for the other students to answer.
Notes about chi-square concepts
- How does the chi-square test work? (measuring how far the observed data are from the null hypothesis by comparing the observed counts to the expected counts - the counts that we would expect to see (instead of the observed ones) had the null hypothesis been true.)
- What is the null and alternative hypotheses for testing two categorical variables? (Ho: There is no relationship between the two categorical variables. (They are independent.); Ha: There is a relationship between the two categorical variables. (They are not independent.))
- How does the chi-square test work? (because we are looking for independence, expected counts are calculated for each cell based on the marginals; the chi-square is the aggregated squared difference between observed and expected counts divided by the expected count)
- What conditions are necessary for the chi-square to be a valid test? (1. the sample is random; 2. expected counts are larger than 5 for all cells)
Visiting research scientist
Shel K. (an engineer at ____)will discuss what role statistics plays in his work and lead an activity related to the design of a research study.
Week 32 (April 29)
Statistics in the big outside world
As students assemble, discuss instances of statistics and probability used in current media.
Discuss concepts in inference for relationships: two independent samples
3 students were asked to develop 2-3 questions based on an assigned set of screens. Have each student present their questions to the other students for the other students to answer.
Notes about concepts
- If the null hypothesis is that all of the means are equal, is Ha then all of the means are not equal? (no, it only takes one mean to be different than the rest to reject Ho...Ha: not all of the means are equal)
- How does the ANOVA F-test differ from the tests we have studied so far? (z and t tests are structured [math]\frac {sample \ stat - null \ value} {standard \ error}[/math]; F-test is structured [math]\frac {variation \ among \ samples} {variation \ within \ samples}[/math])
- What distribution underlies the ANOVA F-test and how do we assess the value of the statistic? (The F-distribution varies for different number of groups and total sample size; a value larger than 4 is usually significant; a p-value is used to determine significance)
- What conditions are required for using F-test? (1) independent samples; (2) response variable is normally distributed in each population, when sample size is small -- check this using the sample data; (3) equal standard deviations for each population -- check this using the sample data, ratio of largest to smallest is less than 2)
Visiting research scientist
Peter S. (an endocrinologist at Johnson and Johnson) will discuss what role statistics plays in his work and lead an activity related to the design of a research study.
Week 31 (April 15)
Statistics in the big outside world
As students assemble, discuss instances of statistics and probability used in current media.
Discuss concepts in inference for relationships: matched pairs
4 students were asked to develop 2-3 questions based on an assigned set of screens. Have each student present their questions to the other students for the other students to answer.
Notes about concepts
- Why bother to create the difference data for calculating the t-test? (allows use of one-sample t-test)
- What do we test in this situation? (mean of sample differences as compared to no difference, Ho; Ha is mean of sample differences is >, <, or =/ to 0)
- How is mu(d) related to mu(1) and mu(2), where mu(d) is the mean of the differences between paired observations in sample 1 and sample 2...x(1) - x(2)? (mu(d) = mu(1) - mu(2))
- What conditions must be met in order to use the matched pairs t-test? (sample of differences is randomly obtained, sample sizes are both large or have normal populations)
- For small samples, how do we confirm that populations are normal?
- What test statistic is used for the matched pairs t-test? ([math]t = \frac {\overline{y_d} - 0}{\frac{s_d}{\sqrt{n}}}[/math])
- How is this test statistic similar to a previous test statistic? (same as one sample t-test)
- Why should Ha be set before doing the study and looking at the data? (the one sided Ha is easier to accept, but would be wrong to set it after seeing the data leans in that direction)
Jeopardy..review of first semester.. for fun
Really...finish the game, including final jeopardy and add-ons. See instructions below.
Decipering research studies
Ask how the students are doing in identifying a study. Discuss strategies.
End-of-course projects
If time discuss ideas for end of course projects.
Week 30 (Apr 8)
Statistics in the big outside world
As students assemble, discuss instances of statistics and probability used in current media.
Discuss concepts in inference for relationships: two independent samples
Before beginning discussion of this week's work, display t-dist vs. z-dist.png to show how the t-dist varies with different sample size and how with larger sample size it approaches the z distribution.
Each student was asked to develop 2-3 questions based on an assigned set of screens. Have each student present their questions to the other students for the other students to answer.
Notes about concepts
- If necessary, display role-type classifications.png.
- As relevant, show graphic Case I - designs.png
- Be sure students understand how explanatory and response variables are used in Ho and Ha. (Ho: Explanatory variable is not related to response; Ha: Explanatory is related to response)
- Conditions under which test can be used? (independence, sample sizes are both large or have normal populations)
- Structure of t-test is similar to structure of previous tests:
- [math]\frac{sample \ estimate - null \ value}{standard \ error}[/math]
- What does a p-value mean? (that in the null hypothesis situation, this is how likely it is to get these results; p-value = .15 means somewhat probable; p-value = .001 means very unlikely; p-value = .061 means ???)
screens 6-7
- What is t* (the multiplier used to create a confidence interval when s is used in place of σ)
- What do we need to use Excel or Calc to calculate margin of error? (alpha, st dev, n)
- How do we calculate the confidence interval when we have two groups???? (if n is large, z* can be used in place of t*, only need to calculate se, show spreadsheet calculations in sleep2.ods)
Jeopardy..review of first semester.. for fun
Finish the game. See instructions below.
Decipering research studies
Additional assignment for next week and week after is to find and summarize (according to the 4 step process) a case 1 research study (or part of a larger study that uses a case 1 test).
End-of-course projects
If time discuss ideas for end of course projects.
Week 29 (Apr 1)
Statistics in the big outside world
As students assemble, discuss instances of statistics and probability used in current media.
Discuss concepts in inference for one variable
- Ask students if there are any concepts that they are confused on. Discuss to help clarify understanding.
- Discuss any concepts that were noted in OLI's "My Response" sections.
Continued...hands-on hypothesis testing for one variable
See Hypothesis testing of a single mean and/or single proportion activity for instructions.
Students completed through data collection for the sleep and language spoken at home scenarios.
- Ask the students to explain to the group what has happened so far.
- Have the students work together to complete the activity.
Jeopardy -- review of first semester...for fun
(still have not gotten to this activity...should be time to begin it this week.)
Play jeopardy with two teams. Use this jeopardy game (note that the Chapt 6 questions need to be connected to the main screen and the answers connected to each problem).
Teams take turns answering questions. If one team answers incorrectly, either team can have a second try--if answered correctly that team gets the points added to their score, if answered incorrectly, the points are subtracted from their score.
- Need to print out the main screen, for use in keeping track of which questions already answered.
- Need to create a list of the answers to use for checking before second try option.
Week 28 (Mar 25)
Statistics in the big outside world
As students assemble, discuss instances of statistics and probability used in current media.
Each student was asked to develop 2-3 questions based on an assigned set of screens. Have each student present their questions to the other students for the other students to answer.
Notes about concepts
Four steps in hypothesis testing:
- State the appropriate null and alternative hypotheses, Ho and Ha.
- Obtain a random sample, collect relevant data, and check whether the data meet the conditions under which the test can be used. (random sample, normal distribution applies--check summary stats) If the conditions are met, summarize the data by a test statistic.
- Find the p-value of the test
- Based on the p-value, decide whether or not the results are significant and draw your conclusions in context.
Screen notes
Screens 9-13 (Hyp Test Pop Prop)
- How does a larger sample size benefit hypothesis testing? Why? (more likely to find significant result, if in fact Ho is false; more evidence, makes stronger case...small margin of error, smaller confidence interval)
- How does statistical significance relate to practical importance? (a larger enough sample size could make any result statistically significant, even one with no practical importance; be suspect of a study with a huge sample size...ask whether the results are practically important)
- Why is the p-value of a two-sided test twice the value of the p-value of a one-sided test? (because the value of the test statistic stays the same, and you have the bit that's greater than the positive value and less than the neg value...twice the amount as a one-sided test, which has a "head start", in a sense, and doesn't need to have as much evidence to reject)
- How does using a confidence interval extend the argument being made in favor of two-sided Ha? (it provides range of data related to the significance value, within which the population parameter does not occur, but also wherein the actual value of p is likely to lie)
- Why is p_{o} used in calculating the z statistic, but p-hat used in calculating the confidence interval? confidence interval is useful when Ho has been rejected, which means p_{o} as the best estimate of the population proportion has also been rejected)
Screens 4-6 (Hyp Test Pop Mean):
- following question on hyp test vs. conf interv, discuss options in ap mc-question #6. key=a
- what is the difference between sd and se in sampling distribution of x-bar? (same formula: sd uses parameter, sigma; se uses statistic, sd)
- study the comparison of the z and t formulations...display z-vs-t.png
Hands-on hypothesis testing for one variable
See Hypothesis testing of a single mean and/or single proportion activity for instructions.
Week 27 (Mar 18)
Statistics in the big outside world
As students assemble, discuss instances of statistics and probability used in current media.
Each student was asked to develop 2-3 questions based on an assigned set of screens. Have each student present their questions to the other students for the other students to answer.
Notes about concepts
Four steps in hypothesis testing:
- State the appropriate null and alternative hypotheses, Ho and Ha.
- Obtain a random sample, collect relevant data, and check whether the data meet the conditions under which the test can be used. (random sample, normal distribution applies) If the conditions are met, summarize the data by a test statistic.
- Find the p-value of the test
- Based on the p-value, decide whether or not the results are significant and draw your conclusions in context.
Four additional points:
- The effect of sample size on hypothesis testing: the larger the sample, the more likely Ho will be rejected (statistic will be determined to be extreme within the Ho distribution)
- Statistical significance vs. practical importance: even tho result may be statistically significant (Ho is rejected), given a sufficiently large sample size you can make any result significant, even one that has very little practical importance.
- One-sided alternative vs. two-sided alternative: it is harder to reject Ho against a two-sided Ha
because the p-value is twice as large; intuitively, the one-sided test gives a head start toward rejecting Ho.
- Hypothesis testing and confidence intervals - how are they related?
Run hypothesis testing demo
Use the instructions at playing cards demo.
Discuss student-created hypothesis testing scenarios
Students were assigned to create and submit a scenario in which hypothesis testing would be useful and Ho and Ha.
For each scenario in turn,
- have the student author present the scenario
- determine what the population is for Ho.
- identify what data we could use to test Ha against Ho
Discuss possible end-of-course project(s)
- Possibilities--real research, real survey
- Why would someone want to do this project? (practice, fun, important,...
- If time, watch A vision of students today -- create something for future stat students to help make the content more relevant!)
- Possibilities--leave a legacy
Jeopardy -- review of first semester...for fun
(didn't get to this last week...good filler if extra time)
Play jeopardy with two teams. Use this jeopardy game (note that the Chapt 6 questions need to be connected to the main screen and the answers connected to each problem).
Teams take turns answering questions. If one team answers incorrectly, either team can have a second try--if answered correctly that team gets the points added to their score, if answered incorrectly, the points are subtracted from their score.
- Need to print out the main screen, for use in keeping track of which questions already answered.
- Need to create a list of the answers to use for checking before second try option.
Week 26 (Mar 11)
Statistics in the big outside world
As students assemble, discuss instances of statistics and probability used in current media.
Confidence intervals for population mean
- What words do we use to explain a specific confidence interval? (We are XX% confident that the unknown population parameter is in the interval (A, B))
- How is a confidence interval created? (point estimate +- z*(σ/sqrt(n)) )
- What are the numbers that are multiplied by the sd of point estimator? (standard deviation units...z*)
- What are three common values for z*? (1.645 for 90% confidennt, 2, or 1.96, for 95% confident, 2.576 for 99% confident)
- Why does the interval get wider as we move from being 90% confident to 99% confident? (need to include more of the possibilities to be more confident)
- What is the tradeoff when deciding how confident we want to be with our interval (higher confidence means less precision....larger range)
- What example might help us understand this? (To be 100% confident, we'd have to include all possible values...which is really unhelpful if the point was to provide data to help understand or make a decision...need to narrow it down a bit...be a little be precise about what we think)
- What is the margin of error? (the amount that is added or subtracted (given a certain level of confidence) to create the confidence interval)
- ....mathematically? ( m = z*(σ/sqrt(n)) )
- how do we use the margin of error? (show image confidence interval structure.png)
- How can we reduce the confidence interval (margin of error) but keep the same level of confidence (get a bigger sample size...reduces sd....reduces m)
- Why does increasing the sample size reduce the margin of error (the sampling distribution is more narrow with a larger sample size, so more sure of point estimate)
- So if it's so easy to be more precise (larger sample) why don't more studies take advantage of this? (cost, availability...)
- What do we do if we want to determine what sample size to use for a given margin of error and level of confidence (solve for n, n = (z* x σ/m)^2 )
- What do we do if we calculate a fractional value for n? why? ( round up; larger sample size is more "conservative" )
- What underlying condition supports the development of a confidence interval? (sampling distribution of estimator, x-bar in this instance, is normal...central limit theorem)
- What conditions must when creating confidence intervals? (the sample must be random, sample is large (n>30) or if sample is smaller that the variable is normally distributed -- show image confidence interval assumptions.png)
- What's wrong with our calculations so far, practically speaking? (σ, the population standard deviation, is often not known!!)
- How do we solve this? (use the t distribution rather than the z distribution to determine the mulitiplier)
- What is the formula for the margin of error when σ is unknown? (substitute sd for σ and t* for z*... m = t*(sd/sqrt(n))
- How is t* different from z*? (depends on sample size as well as confidence level...degrees of freedom)
- What do we call this new formula, in which sd is substituted for σ? (standard error...of x-bar; show SAT score reports, discuss standard error used to create score range, what does it mean, think about the range of scores from one test to the next)
- Under what conditions can we use the sd version of this formula? (same as for pop version...random sample, n>30 unless pop dist is distributed normally)
- When can we use the z*-based calculation to get a pretty good estimate of the t*-based calculation? (large values of n...rule of thumb is >30...interesting discussion of percent error)
Confidence intervals for population proportion p
- In what situations does it make sense to study the population proportion? (categorical variable)
- What is the point estimator for population proportion? (p-hat)
- What is the general form of a confidence interval? (estimate +/- margin of error)
- To create the margin of error, we need a few things:
- multiplier? (z*...1.645, 2, and 2.576 at the 90%, 95% and 99% confidence levels)
- sd of sampling distribution for p-hat? sqrt(p(1-p)/n)... but we don't know p...that's what we are trying to estimate...solution: sqrt(p-hat(1-p-hat)/n)
- What do we call this sd of p-hat when the p-hat is substituted for p? (standard error
- If we want to be more confident that the interval contains the population parameter, what do we have to 'give' on? (precision, narrowness of interval)
- What else can we do to increase precision, for a fixed level of confidence? (increase sample size)
- What practical problem arises when calculating a desired sample size, given a confidence level? (formula uses p-hat, but that's what we want to estimate with the sample...)
- How can we overcome this problem? (use a conservative value for p-hat...one that will make the largest standard error...this is always p-hat=.5...have the students confirm that this is true)
- What is the formula for the conservative estimate of n given m at a 95% confidence level? (n=1/m^2)
- What is the formula for the conservative estimate of m given n at a 95% confidence level? When is this useful? (m=1/sqrt(n); useful to give report convervative result that applies to all questions, even tho quesitons have varying levels of p-hat)
- Under what conditions is it safe to use the methods described to create a confidence interval for pop prop? (p-hat must be distributed normally, therefore n * p-hat >= 10; n(1-p-hat) >= 10)
Discuss and solve student-created confidence interval problems
Students are assigned to create and submit a scenario in which a confidence interval would be useful and to include questions related to creating this interval.
For each problem in turn,
- have the student author present the problem
- discuss and issues and interesting bits
- What is the population?
- What is the sample?
- Is the sample random?
- Is the sample large?
- What do we know about the sample and the population?
- have each student independently answer questions
- share/discuss answers
Discuss possible end-of-course project(s)
- Possibilities--real research, real survey
- Why would someone want to do this project? (practice, fun, important,...
- If time, watch A vision of students today -- create something for future stat students to help make the content more relevant!)
- Possibilities--leave a legacy
Jeopardy -- review of first semester...for fun
(didn't get to this last week...good filler if extra time)
Play jeopardy with two teams. Use this jeopardy game (note that the Chapt 6 questions need to be connected to the main screen and the answers connected to each problem).
Teams take turns answering questions. If one team answers incorrectly, either team can have a second try--if answered correctly that team gets the points added to their score, if answered incorrectly, the points are subtracted from their score.
- Need to print out the main screen, for use in keeping track of which questions already answered.
- Need to create a list of the answers to use for checking before second try option.
Week 25 (Mar 4)
Statistics in the big outside world
As students assemble, discuss instances of statistics and probability used in current media.
- In statistics, what is meant by the term inference (inferring something about the population based on what is measured in the sample)
- What is point estimation? (estimating an unknown population parameter by a single number calculated from the sample data.
- What is a confidence interval? (an estimate of an unknown population parameter by an interval of values, calculated from the sample data, that is likely to contain the true value of that parameter along with an indication of how confident we are that this interval indeed captures the true value of the parameter)
- What is hypothesis testing? (for a stated claim about the population, a decision whether or not the data obtained from the sample provide evidence against this claim.)
- When the variable of interest is categorical, what population parameter do we make inferences about? (population proportion, p)
- When the variable of interest is quantitative, what population parameter do we make inferences about? (population mean, μ)
- In the context of statistical inference, what is an estimator? (a statistic used to estimate a pop parameter)
- ..., what is an estimate? (the value of the statistic that is used as the point estimate for the parameter)
- What do we use as the point estimator for μ? (the sample statistic x-bar)
- Why are the stasticis p-hat and x-bar good estimators for their respective population parameters? (because as long as the sample is taken at random, in the sampling distribution the distribution of the sample mean or sample proportion are exactly centered at the population parameter)
- What does it mean to be an unbiased estimator? (bias is the difference between the expected value of the estimator and the true value of the population parameter being estimated; unbiased means this difference is 0; the estimate is not systematically too low or too high)
- What is required when designing a study to be confident that the resulting estimator is not biased? (sample is random, design is not flawed in some way)
- How can the estimator's accuracy (for predicting parameter) be improved? (larger sample size...the bigger the sample, the more of the pop that's been included, the closer the estimate...larger sample means smaller sd, narrower distribution, better estimate)
- Are p-hat and x-bar the only point estimators? Are there others? (there are lots of others, that's what statistics is all about -- will investigate another point estimator in the upcoming activity; share example of "savings" estimator from local big-box store)
- What's the downside of a point estimator? (It's so often wrong, maybe not by a lot, but enough to make you wonder)
- What can we use to bolster the point estimate? (an interval estimate)
- What does an interval estimate tell us? (the size of the error attached to the point estimate)
- What is the term for an interval estimate? (confidence interval)
- Why is it called that...what two elements are included? (how confident we are that parameter is in the given interval)
- If we want to be 95% confident that the interval contains the population mean, how would we construct the interval (+-2 sd around the mean, with sd being the mean of the sampling distribution)
German Tank Lab
Use the Point estimation - German tank problem activity to investigate the development and use of an estimator for total number in a population.
Jeopardy -- review of first semester...for fun
(snowed out last week)
Play jeopardy with two teams. Use this jeopardy game (note that the Chapt 6 questions need to be connected to the main screen and the answers connected to each problem).
Teams take turns answering questions. If one team answers incorrectly, either team can have a second try--if answered correctly that team gets the points added to their score, if answered incorrectly, the points are subtracted from their score.
- Need to print out the main screen, for use in keeping track of which questions already answered.
- Need to create a list of the answers to use for checking before second try option.
Week 24 (Feb 25)
Statistics in the big outside world
As students assemble, discuss instances of statistics and probability used in current media.
- In a sampling distribution of [math]\bar{X}[/math], what does each value in the distribution represent? (the mean of a random sample from a particular population)
- What else do we need to know about that sample to make sense of the sampling distribution? (how big each sample is: n)
- How many samples are in the distribution? (the assumption is an infinite number....n is NOT the number of samples, but the number of observations in EACH sample)
- What happens if the sample size for all of these means is large, say n=50? (the distribution of [math]\bar{X}[/math] will be normal)
- What if the sample size is small, say n=3? (the distribution may or may not be normal, can't assume so)
- What happens to the standard deviation of the sampling distribution when n is small? (larger) when n is large? (smaller)
- What is the standard deviation of the sampling distribution of the mean? (if you know the population standard deviation, then you can calculate it: [math]\frac {\sigma} {\sqrt{n}}[/math].
- What is the central limit theorem? (it states that given a distribution with a mean μ and variance σ², the sampling distribution of the mean approaches a normal distribution with a mean (μ) and a variance σ²/N as N, the sample size, increases.)
- What is amazing about this theorem? (regardless of the shape of the population distribution, averages that have a large enough sample will have a normal distribution)
- How large must the sample size be for the central limit theorem to kick in? (larger than 30 works for nearly every population distribution; smaller sample sizes will work for pop dists that are somewhat normal.)
- Is there a different sample size rule for when the population distribution has a normal distribution? (Averages based on any sample size will be normally distributed)
Central Limit Theorem problems
Discuss each problem created by the students (additional assignment for last week). What went well? What did they struggle with?
Jeopardy -- review of first semester...for fun
Play jeopardy with two teams. Use this jeopardy game (note that the Chapt 6 questions need to be connected to the main screen and the answers connected to each problem).
Teams take turns answering questions. If one team answers incorrectly, either team can have a second try--if answered correctly that team gets the points added to their score, if answered incorrectly, the points are subtracted from their score.
- Need to print out the main screen, for use in keeping track of which questions already answered.
- Need to create a list of the answers to use for checking before second try option.
Week 23 (Feb 18)
Statistics in the big outside world
As students assemble, discuss instances of statistics and probability used in current media.
Useful Content?
- Do you think it's worth learning about how to use a normal random variable to approximate probabilities of a binomial random variable (covered in the last 2 screens of the normal random variables section)? Why or Why not?
[My opinion is that binomial probabilities can be calculated quickly and easily with a scientific calculator, a spreadsheet or a web-based binomial calculator. No need to use the normal distribution as an approximation for calculating probabilities.]
- What is sampling variability? {the idea that the characteristics of a sample, from a given population, vary from one sample to another)
- What is a parameter? (a number that describes the population)
- What is a statistic? (a number that is computed from the sample)
- What symbol is used to denote a population proportion? ([math]p[/math]) a sample proportion? ([math]\hat{p}[/math])
- What symbol is used to denote a population mean? (μ) a sample mean? ([math]\bar{X}[/math])
- What symbol is used to denote a population standard deviation? (σ) a sample standard deviation? (s, or sd)
- Why are parameters generally unknown, as compared to the ease of calculating statistics? (usually impractical or impossible to know the value of a variable for every individual in the population)
- What is the sampling distribution of [math]\hat{p}[/math]? {the distribution of the values of the sample proportions [math]\hat{p}[/math] for a large number of repeated samples)
- What does this distribution look like (it depends on the size of the samples; for large enough samples it can be approximated by a normal distribution)
- What is the mean of this sampling distribution? (p)
- What is the standard deviation? [math]\sqrt{\frac{p(1-p)}{n}}[/math]
- How large does the sample size need to be such that it's appropriate to use the normal distribution to approximate the sampling distribution? (np and n(1-p) are at least 10)
- How does the mean of the sampling distribution vary with larger samples? (the distribution of [math]\hat{p}[/math] is always centered at p, but the larger the sample size, the better [math]\hat{p}[/math] estimates [math]p[/math])
- How does the spread of the sampling distribution vary with larger samples? (the larger the sample, the smaller the spread of the sampling distribution)
Sample proportions
Have students describe their two instances of sample proportions (including description of population) from the news/current media.
Standard normal calculations: Are these M&M's "normal" or just "plain"
Have each student do calculations independently (using normal table, spreadsheet or graphing calculator), noting answers aloud as you go.
Materials:
- 2 bags of 1.69oz milk chocolate M&M's
- 1 bag of almond (or peanut butter) M&M's
- graphing calculator, normal table or spreadsheet software for each student
- paper and pencil
Introduction
According to the M&M website, 14% of milk chocolate M&M's are yellow.
- Does this mean that in every bag 14% of the candies are yellow?
- Is it reasonable for a bag to contain only 10% yellow?
- What could we do to convince ourselves that the advertised proportion is correct?
Let's think about this as a sampling distribution of [math]\hat{p}[/math] compared to the population proportion, [math]p[/math].
- What represents each sample (bags of milk chocolate M&M's)
- What is the population (all milk chocolate M&M's)
In response to question #1, our experience with the color variability across bags of M&M's suggests that the number of yellow M&M's varies from one bag to the next. Let's assume that the distribution of proportions follows a normal distribution: N(.14, .05).
- Sketch the distribution and note 1, 2, and 3 standard deviations above and below the mean.
- Interpret this distribution using the standard deviation rule. (68-95-99.7)
Empirical evidence
Count the number of yellow M&M's and the total number of M&M's in a 1.69oz bag of milk chocolate M&M's.
- Is it reasonable to use a normal distribution to model the proportion of yellow M&M's in a bag? (np and n(1-p) are both >= 10)
- What is the percent of yellow M&M's?
- What is the z-score for this observed value? (mean-value/sd)
- Draw two normal distributions.
- Determine the probability of obtaining a proportion of yellow M&M's that is less than the observed value; shade this area on your sketch.
- Determine the probability of obtaining a proportion of yellow M&M's that is more than the observed value; shade this area on your sketch.
Compute the percent of yellow M&M's from a second 1.69oz bag of milk chocolate M&M's.
- How does this compare with the first value.
- What is the probability of obtaining a proportion between these two values?
What about peanut butter (or almond) M&M's
Do peanut butter m&m’s follow the same color distribution as milk chocolate m&m’s? If so, we would expect about 14% of the candies in a peanut butter m&m bag would be yellow. Would you be surprised if 15% were yellow? What about 20%? 30%? At what point would you suspect the color distribution for peanut butter m&m’s may be different?
- Open 2 bags of peanut butter M&M's.
- Calculate the proportion of yellow M&M's.
- Plot each value on a normal distribution with mean=.14 and sd=.05 (the milk chocolate population)
- For each bag, determine the probability of a value more extreme than this.
Discuss: Based on this %, do you feel you have evidence to suggest the color distribution of peanut butter M&M’s may be different? Why or why not?
adapted from Standard Normal Calculations at StatsMonkey
Creating a sample distribution
Let's consider the counts by color for 66 fun-size bags of M&M's in the M&M spreadsheet that you created in the first few weeks of this course to represent a sample distribution of [math]\hat{p}[/math] for each color.
Issue with using bags of M&M's to represent independent samples in a sampling distribution from the population (N varies from bag to bag).
Using the spreadsheet do the following:
- Convert the count for each bag into a proportion.
- Calculate the mean of [math]\hat{p}[/math]. (to estimate the population proportion)
- Calculate the standard deviation of [math]\hat{p}[/math].
- Plot the distribution of [math]\hat{p}[/math] as a histogram.
Discuss the shape of the distribution. Is it normal looking? If not, consider reasons why it might not be.
Proportion of M&M's by color, provided in a letter from Mars Snackfood US:
- M&M'S MILK CHOCOLATE: 24% cyan blue, 20% orange, 16% green, 14% bright yellow, 13% red, 13% brown.
- M&M'S PEANUT: 23% cyan blue, 23% orange, 15% green, 15% bright yellow, 12% red, 12% brown.
- M&M'S PEANUT BUTTER and ALMOND: 20% cyan blue, 20% orange, 20% green, 20% bright yellow, 10% red, 10% brown.
Week 22 (Feb 11)
- What is a continuous random variable? (a variable that can take on any value in a given range)
- What is a probability density function? (a formula that can be used to compute the probabilities of a range of outcomes for a continuous random variable)
- How is a probability density function different from a histogram displaying the probabilities of a discrete random variable? (total area under the curve = 1 vs. sum of heights = 1)
- How do you use a probability density function to determine probabilities? (determine the appropriate area under the curve).
- What is the probability of a particular single outcome? (zero, because there is no area associated with a single value, the exact value to many decimal places--there are an infinite number of values for any continuous random variable)
- When does it matter if we write P(X<=3) vs. P(X<3)? Why? (Need to specify whether the value is included or not for a discrete variable; for a continuous variable, can specify up to the edge of the value because no area associated with a particular value)
- What shape is a density curve? (any shape--only restriction is the area under the curve = 1)
- How can we find a specified area under a probability density curve? (calculus, specifically integration; we will use tables and spreadsheet functions)
- What is a normal distribution? (a bell-shaped probability density curve that can describe many natural phenomena)
- What is the standard deviation rule (percentages of observations distributed in a bell-shaped curve representing 1, 2, and 3 standard deviations out from the mean: 68, 95. and 99.7 %s)
- How does the standard deviation rule (learned in the first part of this course) relate to our understanding of a normal distribution as a probability density curve? (68% of observations fall between +-1 sd of the mean becomes P(μ-σ < X < μ+σ) = .68)
- What do we do if we are interested in probabilities that aren't related to the sd? (need a way to estimate the area under the curve)
- What does it mean to standard a normal value? (determine how many standard deviations it is away from the mean; (x-μ)/σ)
- What is this standardized normal value called? (z-score, a value of the standard normal random variable Z)
- What kind of values will a z-score take? (z-scores from about -3.5 to +3.5 correspond to >0 probabilities)
- What is a normal table? (a table of z-scores listing the probability of obtaining a value less than, to the left of, that value)
Show a normal table with *.* in the rows and .** in the columns.
- What is the probability of a normal random variable taking a value less an 1.33 standard deviations above the mean? ( .9082)
- What are the two methods for solving a problem that asks for the probability of a value greater than the z-score? (1. use the symmetry of the table and solve for less than the negative of the value; 2. find the probability for less than the value and then subtract from 1)
- How can you solve for probabilities between two values? (Find the probabilities below each value and subtract the smaller one from the larger)
Note that a sketch will often help you keep the calculations straight.
- How do we find the corresponding number of standard deviations away from the mean when given a probability value describing an area of the curve above, below or between? (for below, find the probability value in the table and note the corresponding z-score, for above and between use the symmetric or adds-to-1 properties of the normal distribution to transform the probability in a less than value.)
- What information do you need to calculate the probability of an observation greater than some value? (use mean and sd of distribution to convert value to z-score--compute probability; also need some assurance that using a normal distribution is appropriate)
- How do we notate these transformations? (P(X > 13) = P(Z > 1.33) = P(Z < -1.33 = .0918)
Feel comfortable working from probability through z-score to observed value?
- Besides proportion or percent, what other term is used to express probabilities in a normal distribution (percentile, 25 % score below (or at or below) the 25th percentile)
Approximating the binomial distribution
We can use the normal distribution to approximate the binomial distribution.
- Why is this useful? (probabilities in a binomial are a pain to calculate)
- Do you think this is still an important methodology given the statistical tools available? (note that interpolation, estimating for a probability between two values in the normal table, is no longer needed, given statistical tools)
- How is the approximate normal distribution created from the binomial? (X_{N} has mean = np, and sd = sqrt(np(1-p)))
- What is the rule of thumb for having confidence in the normal approximation? (np >= 10 AND n(1-p) >=10)
- How does the continuity correction work (use the value that is halfway between the two relevant binomial values as the value in the normal distribution computations)
Review of Random Variables module
Group students in groups of 2-3. Have each group develop 5 questions related to content presented in the random variables section. Have groups take turns asking and answering questions, no notes allowed (can use scrap paper for calculations).
Week 21 (Feb 4)
Binomial RV problems
How'd it go? Did get done as easily as ones previously. Wondering why?
- What are the requirements for a random experiment to be a binomial experiment?(1. a fixed number n of trials; 2. each trial must be independent of the others; 3. each trial has just two possible outcomes, called "success" (the outcome of interest) and "failure"; 4. there is a constant probability p of success for each trial (and so the probability of failure is (1-p)))
Have each student give an example of a binomial random experiment.
- Discuss the following example from OLI (binomial random variables 1 of 4); OLI says the example is not binomial. Do we agree?:
- (f) Approximately one in every 20 children has a certain disease. Let X be the number of children with the disease out of a random sample of 100 children. Although the children are sampled without replacement, it is assumed that we are sampling from such a vast population that the selections are virtually independent. We can say that X is binomial with n=100 and p=1/20=.05. The experiment is not binomial because there are more than two possible outcomes for each trial.
- What is the sample size requirement for a sample's independence to be unaffected by non-replacement? (pop is 10 times the sample size.)
Binomial formula:
- What are the different terms in the formula? (each term represents the probability for x successes (our of n trials).
- What are the two parts of each term? (a count of the number of ways to achieve the number of successes multiplied by the probability of success x times and the probability of failure n - x times)
- How can we determine the count of the number of ways to achieve x successes? (formula=n!/x!(n-x)!; also called n choose x, written _{n}C_{x}) -- the Khan Academy video provides an excellent intuitive understanding of this concept.
- What does 5! mean? (5*4*3*2*1=120)
- What is 0!? (1)
- What is the mean of a binomial random variable? (μ_{X}=np)
- What is the sd of a binomial random variable? (σ_{X}=[math]\sqrt{np(1-p)}[/math]; 1-p is also called q)
- Are the mean and sd of a binomial different than for other discreet random variables? (the general rule works also, but easier to use these formulas.
Next week
Lots of work on the Normal distribution. This is an extremely important concept. Time invested now to fully understand how to apply the normal distribution will definitely pay off later. No additional assignment.
Next week's meeting -- Feb 11.
- 10-11:30am?
- preferred location?
Lucky dice lab
Show the images chuck-a-luck.jpg and Bau_cua_ca_cop.jpg.
Group the students in groups of 2-3. Have each group complete the Lucky dice lab). Need 3 dice for each group.
As the groups work on the lab, google "chuck a luck" and play "the legend of chuck a luck" song from Rancho Notorious.
Week 20 (Jan 28)
Watch Khan Academy video on random variable, from beginning to 2:30 min.
- Anything helpful in Salman's explanation?
Discussion of OLI content
- What is another term for the mean of a random variable? (expected value)
- OLI chooses not to use the term expected value when referring to the mean of a random variable. Why? (For some distributions the expected value is not a possible outcome, so how is it the most expected value)
- Why is the mean of a random variable denoted as μ_{X} rather than X-bar? (mean of the population)
- Describe how to calculate the mean of a random variable using the probability distribution.
- How can we apply the concept of a probability dist to calculate the mean of a sample? (for each value in the sample, determine its prob of occurring, use the probdist to calc mean)
- What is the standard deviation of a random variable and how is it denoted? (average deviation from the mean denoted σ_{X})
- Describe how to calculate it (subtract each value from the mean, square it, multiply by probability and add them all together, take the square root)
- Why is the deviation squared in the calculation?
- What is the variance of a random variable (the average squared deviation)
- How does adding a constant affect the mean and standard deviation of a probability distribution (Mean + constant = new mean, sd is unchanged)
- How does multiplying a constant affect the mean and standard deviation of a probability distribution (both are multiplied by the constant -- relocates and expands or contracts the distribution)
- How does multiplying a constant affect the variance? (constant^2 times the original variance)
- How can you summarize both of these changes into one concept? (linear transformation a+bμ, b^2*σ^2)
- What is the mean of a random variable that is the sum of two random variables? (meanX + meanY)
- What is the standard deviation of a random variable that is the sum of two random variables? (sqrt of σX^2 + σY^2)
- On what condition? (independence)
Coins, dice, cards lab
Group the students in groups of 2-3. Have each group complete the Coins, Dice, Cards lab (station 5 in CasinoLab.pdf). Will need 4 coins, pair of dice, and deck of cards for each group.
Extra time: Begin Binomial Distribution
Watch the Khan Academy videos:
- Binomial Distribution 1, 12:15 min
- Binomial Distribution 2, 11:05 min
Week 19 (Jan 21)
- What is a random variable? (a variable whose values are numerical results of a random experiment)
- What are the two kinds of random variables that we care about? (discrete and continuous)
- What characterizes a discrete random variable? (possible values are a list of distinct values)
- What are examples of discrete random variables in disguise? (rounded values)
- What are examples of discrete random variables that are better dealt with as continuous variables? (a variable with a lot of possible values--test scores)
- How do we write the probability of a particular value of the random variable? (P(X=x)
- What is a probability distribution? (The collection of values and probabilities associated with a particular random variable)
- What properties must a probability distribution of a random variable fulfill? (1. each probability is bwtn 0 and 1, inclusive, 2. the sum of the probabilities =1)
- What tool can we use to summarize a probability distribution? (a table -- show and discuss image of table from OLI, probdist table.png)
- How is a probability histogram different from the distribution histogram we talked about earlier (y axis represents probabilities)
- Display obama election results probdist.png - discuss how this graph is a probability distribution (random variable is results of state polls; y-axis is in percent not proportion; seems like it could add to 1; discrete altho a lot of values)
Problems
Discuss "Sales call" OLI problem (given Tim's concern that it is illogical).
(from Collaborative Statistics, Discrete Random Variables Homework
Exercise 2. Suppose that you are offered the following “deal.” You roll a die. If you roll a 6, you win $10. If you roll a 4 or 5, you win $5. If you roll a 1, 2, or 3, you pay $6.
- a. What are you ultimately interested in here (the value of the roll or the money you win)?
- b. In words, define the Random Variable X.
- c. List the values that X may take on: x=....
- d. Construct a probability distribution.
- e. Over the long run of playing this game, what are your expected average winnings per game? (this question foreshadows future learning of expected value--mean of the prob dist)
- f. Based on numerical values, should you take the deal?
Exercise 6. Suppose that the probability distribution for the number of years it takes to earn a Bachelor of Science (B.S.) degree is as shown in Bachelor degree year probdist.png.
- a. In words, define the Random Variable X
- b. What does it mean that the values 0, 1, and 2 are not included for X on the PDF?
- c. What is the probability of completing a bachelors degree in at most 4 years?
- d. Of those that take longer than 4 years, what is the probability of completing in more than 6 years?
Conditional probability lab
Continuation from last week, with specific focus on creating a computer simulation of a large number of trials:
Calculate a selection of probabilities based on the contents of a regular bag of m&ms (about 40). Use lab directions at Connexion's Probability Topics: M&M Lab.
Week 18 (Jan 14)
Review of concepts in probability
- What is the General Multiplication Rule? (P(A and B)=P(A)*P(B|A))
- Can the general rule be use with independent events? Why, what happens?
- How can we understand the general multiplication rule conceptually? (solve conditional probability formula for P(A and B), or realize that second probability is dependent on the first--uses the conditional probability)
- What is the Law of Total Probability? (P(B)=P(A and B) + P(not A and B)=P(A)*P(B|A) + P(not A)*P(B|notA)
- How can we understand the Law of Total Probability conceptually? (Here's how to talk about it: If I want to calculate the probability of B, if A happens (with P(A), then B occurs with probability P(B|A) and if A doesn't happen (P(not A), then B occurs with probability P(B|not A); also suggest a Venn diagram where A, not A describe the universe and B occurs within that universe)
- What can we use these two rules to reformulate the equation for P(A|B)? (display the 3 equation derivation from the bottom of this OLI page)
- When is it useful to use this formulation? (when you know more about the second event than the first.)
Walk-through the slideshow Probability Trees and Bayes' Theorem to reinforce conceptual understanding.
Conditional probability lab
Calculate a selection of probabilities based on the contents of a regular bag of m&ms (about 40). Use lab directions at Connexion's Probability Topics: M&M Lab.
Week 17 (Jan 7)
Review of concepts in probability
Discuss the meaning and nuance of the following:
- What does conditional probability mean? (the probability of a second event conditioned on a prior event)
- How do we interpret P(B|A)? (the likelihood that a chosen A is also B; probability of B given A)
- If you had a two-way table of counts for A, not A and B, not B, how would you calculate P(E|M)?
- What is the formula, using other probabilities, for calculating P(B|A)? (P(B|A) = P(A and B)/P(A))
- How does this formula reduce to the same formula used in the calculation based on the two-way table? (denominators in both probability fractions are the same, and cancel out leaving the counts from the two-way table)
- When is the conditional probability formula undefined and how should we think about that situation? (when P(A) = 0; can't find a probability of something given an impossible event)
- How can we implement the complement rule with conditional probabilities? (most important to use complement ONLY when conditioned on same event P(B|A) = 1 - P(not B|A))
- What does it mean for 2 events to be independent?
- How can we use the conditional probability to check whether two events are independent? (independent if P(A|B) = P(A))
- Can we use P(A|B) and the P(A|not B) to check for independence? Why or why not? (Yes, because if they are the same, then whether or not B happens is irrelevant, so they events are independent.)
- How can we use the multiplication rule to check for independence? (If P(A and B), e,g, computed from a two-way table, equals P(A) * P(B), then events A and B are independent)
Moodle
Discuss how the stats course moodle can be used to enhance the course.
Group problem solving
Instruct the students to work each probability problem as presented. Then ask a student to present and explain the answer. Use the following problems from Collaborative Statistics' probability homework:
#1, answers-- a. {G1, G2, G3, G4, G5, Y1, Y2, Y3} b. 5/8 c. 2/3 d. 2/8 e. 6/8 f. No, P(G and E)is not equal to 0
#2, answers-- a. skip b. 5/8 * 5/8 = 25/64 c. 1-P(Y1 and Y2) = 1 - (3/8 * 3/8) = 55/64 d. P(G1 and G2)/P(G1) = 25/64 / 5/8 = 5/8; of course it's equal to P(G1) because the events are independent e. yes, because P(G2|G1) = P(G2), so it doesn't matter whether or not G1 occurred
#4, answers-- a.
1 | 2 | 3 | 4 | 5 | 6 | |
---|---|---|---|---|---|---|
1 | 1,1 | 1,2 | 1,3 | 1,4 | 1,5 | 1,6 |
2 | 2,1 | 2,2 | 2,3 | 2,4 | 2,5 | 2,6 |
3 | 3,1 | 3,2 | 3,3 | 3,4 | 3,5 | 3,6 |
4 | 4,1 | 4,2 | 4,3 | 4,4 | 4,5 | 4,6 |
5 | 5,1 | 5,2 | 5,3 | 5,4 | 5,5 | 5,6 |
6 | 6,1 | 6,2 | 6.3 | 6,4 | 6,5 | 6,6 |
b. P(A) = 2/6 * 3/6 = 6/36 = 1/6 c. P(B) = 21/36 d. P(A|B) = P(A and B)/P(B) = 3/36 / 21/36 = 3/21 = 1/7 e. no, both and A and B can occur > 3,4; P(A and B)=1/36 f. no, knowing whether B occurred or not affects the probability of A. P(A|B) is not equal to P(A).
#8, answers-- a. P(C and D) = P(C|D) * P(D) = .6 * .5 = .3 b. skip c. no 1) P(C|D) not equal to P(C); 2)P(C) * P(D) = .2 which is not equal to P(C and D) 3)could we calculate P(C|not D) to compare to P(C|D), suggest creating the two-way table. d. P(D|C) = P(D and C) / P(C) = .3 / .4 = .75
#10, answer-- P(J) = .3 because the events are independent. What happened with K doesn't matter.
#19, answers-- a. iii b. i c. iv d. ii
Week 16 (Dec 17)
Review of concepts in probability
Discuss the meaning and nuance of each of the following rules, using these questions:
Prep for Rule 5:
- What does P(A and B) mean? Describe a Venn diagram that demonstrates this statement.
- What is P(A and B) if A & B are disjoint? (0)
- What does it mean for two events to be independent? (Two events A and B are said to be independent if knowing whether one event has occurred does not affect the probability that the other event occurs.)
- What is the converse of independent? (dependent)
Have each student provide an example of two independent and two dependent events. For each, specify whether the events are disjoint or not disjoint.
- Movivating example: P(person saw the smile commercial) and P(person signs up with AmEx credit card) - dependent.
Have 1-2 students describe two events that are disjoint -- discuss whether they are independent or dependent. (always dependent)
- Rule 5: The Multiplication Rule for Independent Events: If A and B are two independent events, then P(A AND B) = P(A) * P(B) (Must be careful to use this only for independent events; discuss how joint probability would be effected for dependent events identified earlier.)
- What does it mean to determine the P(at least one of). What is the difficulty of solving this kind of problem (prob of 1 + prob of 2 + ... + prob of all)? Is there a way around this? (complement rule)
- In the statement "what is the probability of at least 1 of 10 monkeys scoring in the video game (by chance)", what is the complement? (none of the monkey's score)
- The probability that a monkey will score is .1. How do we use the complement rule to calculate the prob of at least 1 monkey scoring? (1-.9^10 = .65)
Rule 6: The General Addition Rule: For any two events A and B, P(A or B) = P(A) + P(B) - P(A and B) (Is there anything that you need to be aware of when using this formula? Be careful to only multiply P(A) and P(B) to get P(A and B) when A and B are independent. What if A and B are dependent? Need to obtain P(A and B) by observation.)
Is there an error in "Learn by Doing" on page 10? Anyone figure out what is wrong?
Discuss probability in games of chance
Continue discussion of games (as described for week 15)
Also discuss idea of ordered and unordered outcomes for dice (what the heck is the difference and why does it matter?)
Randomness as explained by Charlie on NUMB3RS
- Additional resource: Double Down
Ask the students to spread themselves into a random pattern.
While standing in their spots, watch the "what's random" segment from the pilot of Numb3rs, at 17:50.
Have each student flip a coin 14 times and record the outcome. Ask each student to read out their sequence and tell how many H and how many T. Ask the students to compare their result with getting exactly HTHT, or all H's or all T's.
- Which is more/less likely?
- What is the probability of any one of these sequences? (all the same - .5^14)
- How does this compare with what Charlie is telling us.
- What do people do when they want something to appear random? Why do you think people think "spread evenly" when they want random?
Are we coins? from NPRs Radiolab
Discuss the concept of streaks in sports (or in gambling). How can we understand streaks in the context of an event having a given probability of occurring.
Play this radio essay, "Are We Coins" from NPR's Radiolab, about probability and simulations and randomness.
Course evaluation
How is the course working out for each of you?
- Too much? Too little?
- OLI coursework?
- Thursday sessions?
Any changes you would suggest to encourage better learning?
Week 15 (Dec 10)
Review of concepts in probability
Discuss the meaning and nuance of each of the following rules
- Rule 1: For any event A, 0 ≤ P(A) ≤ 1 (if a probability is greater than 1, something is wrong)
- Rule 2: P(S)=1; that is, the sum of the probabilities of all possible outcomes is 1. (we can figure out missing probabilities; if the sum of the probabilities in the sample space is greater than 1, something is wrong)
- Rule 3: The Complement Rule: P(not A) = 1 - P(A) (sometimes it's lots easier to calculate when something doesn't occur than when it does.
- What is the compliment of the birthday problem? (everyone in the group has a different birthday)
Prep for rule 4
- What does 'OR' mean in probability?
- What does disjoint mean? (2 events that cannot both occur at the same time) How else might it be termed? (mutually exclusive)
Have each student provide an example of two disjoint events, and two non-disjoint events.
- Rule 4: The Addition Rule for Disjoint Events: If A and B are two disjoint events, then P(A or B) = P(A) + P(B) (why do you think this rule only applies to disjoint events? (draw a picture of non-disjoint events, overlapping circles, to motivate understanding)
Discuss probability in games of chance
Each student presents about the chance game they investigated. The following should be identified:
- random experiment in the game (such as winning/losing a round or hand)
- sample space with enumerated outcomes
- determine the probability of each outcome or a selection of outcomes
After the presentation, play a round or two of the game.
What assumptions are integral to the validity of the probabilities? (e.g., cards are randomly distributed in the deck -- btw, requires at least 5 shuffles and 7 to be sure)
Simulating guessing on a true-false test
See instructions for true-false test activity here.
This week's work
Online work:
- Unit 4: Probability
- Mod 6: Finding Probability of Events
- Probability Rules - Investigate Difference between Disjoint and Independent Events; Find P(of at least one of...); Find P(A or B)
- Mod 6: Finding Probability of Events
For our next session:
Week 14 (Dec 3)
Review missed quiz questions
Display questions 8 (read stim first), 10, 14, and 18; compare answer to distractors
- For Q8, identify population, and sample and then sampling frame.
- For 14, discuss idea of randomized response
- For 18, read through answer provided; discuss ideas for grading rubric
Statistics discussions
Each student summarizes their "adult" discussion relating to statistics.
Review of concepts in probability
- How intuitive are we at guessing probability? (display BD response graph.png)
- What is a random experiment (an experiment with an unknown outcome)
- What is an outcome? (the situation that results from an experiment)
- What does the capital letter S stand for (sample space -- enumeration of possible outcomes)
- What's an event? (a statement describing a collection of outcomes in the sample space)
- What's the probability of an event? (a number that tells us how likely an event is to occur)
- How are probabilities expressed? (as proportions)
Have each student describe a random experiment, identify its sample space, and name a relevant event (give them 2+ min to prepare). Decide whether order matters in the outcomes. Can we calculate the probability of the event? Why or why not? (discuss connection to ideas of relative frequency and equally likely).
- What is relative frequency (empirical probability, measure the proportion of times event occurs in a large number of trials)
- What did people calculate as the relative frequency of having 2 or more of the same birthdays at a party of 30 people? (one .7, two .9, one .5)
- What does it mean for outcomes to be equally likely? In what kind of situations does this occur? What're the outcomes in a basketball free throw? (success failure) Are they equally likely? (depends on player's skill)
Have each student choose a marble from a bag containing different sized marbles. Does each marble have the same chance of being chosen?
Data simulation: Counting successes
Use the activity on p. 220 (Activity C1, on the page numbered 206) at [1]. Rather than require use of a random number table, have the students design their own random number generator (encourage use of a spreadsheet program or a TI-83+ calculator).
Discuss questions b.i. and b.ii. as a group
This week's work
Online work:
- Unit 4: Probability
- Mod 6: Finding Probability of Events
- Probability Rules - Investigate Disjoint vs. Not Disjoint Events; Investigate Application of Addition Rule
- Mod 6: Finding Probability of Events
For our next session:
- Probability and games of chance: choose a game such that winning/losing is all (or nearly all) chance. Choose something simple. Study the game to identify the "experiment" and what the outcomes are. Determine the sample space, and the probabilities for each (or a selection of) outcomes. Determine the probabilities for the event "winning" and the event "losing". I'd like each of you to have a different game, so you will email the group with what game you choose. Fist come, first served.
Week 12 (Nov 19)
Questions about Producing Data
- Different kinds of randomness
- Names of the different types of surveys and studies (use your notes)
Summaries of real research reports
Each student presents on the research report that they read:
- the question asked in the study (If there's more than one, just choose one.)
- the explanatory and response variables
- how the sample was selected (method and sampling plan)
- the study design (experimental, observational, or survey)
- the treatment groups, if any
- the outcome
- a causal diagram
Some example studies for discussion
carried over from Week 11
- Suppose two researchers wanted to determine if aspirin reduced the chance of a heart attack. Researcher 1 studied the medical records of 500 patients. For each patient, he recorded whether the person took aspirin every day and if the person had ever had a heart attack. Then he reported the percentage of heart attacks for the patients who took aspirin every day and for those who did not take aspirin every day.
- Researcher 2 also studied 500 people. He randomly assigned half of the patients to take aspirin every day and the other half to take a placebo everyday. After a certain length of time, he reported the percentage of heart attacks for the patients who took aspirin every day and for those who did not take aspirin every day. Suppose that both researchers found that there is a statistically significant difference in the heart attack rates for the aspirin users and the non-aspirin users and that aspirin users had a lower rate of heart attacks.
- What is the design of each study? Can researcher 1 conclude that aspirin caused the reduction in rate of heart attacks? Why or why not?
- Source: SOCR Problem 2
- Hospital floors are usually covered by bare tiles. Carpets would cut down on noise but might be more likely to harbor germs. To study this possibility, investigators randomly assigned 8 of 16 available hospital rooms to have carpet installed. The others were left bare. Later, air from each room was pumped over a dish of agar. The dish was incubated for a fixed period, and the number of bacteria colonies were counted.
- What is the appropriate statistical term for the 8 rooms left bare. (Control group)
- What do we call the 16 rooms that were chosen for the study (sample)
- What do we call the number of colonies growing in each dish (Response variable)
- Source: SOCR Problem 4 and 7
- As I read the following article decide what you think are explan and response variables, subjects, and study design:
- Flexible Brains People who grow up left-handed have a different, more flexible brain structure than those born to take life by the right hand, says UCLA researchers who use twins to study heredity. The reason is that right-handers have genes that force their brains into a slightly more one-sided structure. Left-handers appear to be missing those genes.
- “There is a real difference in brains that result in a more symmetric brain in left-handers, where the two sides are more equal,” said UCLA neurogeneticist Dr. Daniel Geschwind, who lead the research team. “There is more flexibility and that is under genetic control.” That hereditary difference between right-handers and left-handers also appears to affect how the brain changes in size throughout a lifetime, the researchers found.
- Of all the primates, only humans display such a strong predisposition to right-handedness. Right-handers make-up about 90% of the population. The left and right halves of the brain are different in both anatomy and their features, related to hand preference. But until now, no one could document the connection.
- To study brain size and structure, the UCLA researchers used brain-scanning technique called functional magnetic resonance to compare brains in 72 pairs of identical twins, all of them male World War II veterans ages 75 to 85. Identical twins-who share the same genes- offer a unique lens through which to study the relative effects of heredity on human nature.
- Right-handers typically have a larger left brain hemisphere, where their language abilities are concentrated. Conversely, left-handers have more balances brains, with both sides relatively symmetric. “Overall, this study shows us that brain structure is highly influenced by genetics, even later in life,” Greschwind said. “This implies that aging-related changes to the brain also possess a strong genetic basis. That is kind of wild.”
- What is explanatory variable? (left and right handedness)
- What is response variable? (brain size and structure)
- Who are the subjects? (72 pairs of male twins who were WWII veterans, ages 75 to 85 years)
- What is the study design? (observational)
- Is this a matched pairs design? (not really, cause it's not an experiment; but the twins control for lurking variables to some extent)
- What can be concluded about causation?
- Source: SOCR Problems 12-15
Work covering next two weeks
Online work:
- Unit 3: Producing Data: quiz and summary
- Begin Unit 4: Probability
- Mod 5: Introduction (Probability)
- Mod 6: Finding Probability of Events
- Relative Frequency - Calculate Relative Frequency
- Equally Likely Outcomes - Calculate Probability of Event
For our next session:
- Possibly send something out after Thanksgiving.
Design an experiment
carried over from Week 11
Groups collaborate to design an experiment to address a research question. Work together to decide on the following:
- Research question
- Explanatory variable
- Values?
- Control group?
- Response variable
- How is it measured?
- Study design
- Sampling plan
Week 11 (Nov 12)
Review quiz questions
Display questions 1, 5, and 10 and discuss distractors vs. correct answer.
- For #5, show nonlinear high r.png example.
- For #10, discuss hospital example, higher death rate in A compared to B, but when break out severely ill from moderately ill, B has higher death rate in both. IT'S A PARADOX.
Review terminology in study design
- Why is a correlational study design usually observational?
- What is an experimental factor?
- What does ttt represent?
- What is a control group?
- What are the subjects?
- How does random assignment to treatments contribute to establishing a case for causation?
- What kind of experiment results? (randomized controlled experiment -- examplanatory variables are controlled)
- Does an experiment have to have a control group? (No) What are some examples?
- How can lack of a control group contribute to a flawed experiment? (no comparison) Example?
- Three uses of the concept of control? (controlled observational study -- break out results to control for lurking variable; controlled experiment -- researchers assign values of explan var, treatment, rather than occurring naturally; control group -- subjects who do not receive treatment, but are the same in every other respect)
- What does blind mean when referring to an experiment?
- What's a placebo? (I will be pleased)
- What's the placebo effect? (idea that subject will feel better just by thinking he's being treated)
- What is the experimenter effect? (researcher unconsciously -- or consciously -- influences outcome)
- How can an experiment be designed to avoid this problem.
Read the Clever Hans Story, discuss application to research design.
- What is the Hawthorne effect?
Play the first bit of Hawthorne Effect radio show, from the BBC, for at least 8 minutes to get the results of relay studies.
- What is the problem of lack of realism (lack of ecological validity) in experiments?
- How can non-compliance effect an experiment? (subjects fail to complete assigned treatment, could be due to some systematic reason) How can an experiment minimize non-compliance (volunteers; pay)
- Where is randomization most important?
- When is assigning treatment groups not approp? Give some examples (illegal drugs, physical/cultural/social characteristics)
- How is an experiment with more than one explan var designed (full crossing of values in each variable, factorial experiments)
- How can the experimenter randomly assign the same number of subjects to each group? (repeated SMS of desired group size)
- What is blocking? How is it different from statification? (assignment of subjects vs. sampling)
- What is the matched pairs design?
- What are the variations of matched pairs? (two treatments per individual, matched individuals assigned to separate treatments, before-after studies)
- How should randomization be included for each variation? (random order vs. random assignment vs. can't)
- What naturally occurring group is used often in matched pairs experiments?
Turning to a discussion of surveys.
- What is an open question? Why is this a problem in a survey?
- What needs to be considered when writing a closed question (all options are understandable/represented/offers option to not say, or other)
- What is meant by unbalanced response options? (more +/- responses, no middle choice)
- What are leading questions? example?
- Why is the order of questions important?
- How is the idea of randomized response used to obtain honest response to sensitive question (coin flip)
Some example studies for discussion
- Suppose two researchers wanted to determine if aspirin reduced the chance of a heart attack. Researcher 1 studied the medical records of 500 patients. For each patient, he recorded whether the person took aspirin every day and if the person had ever had a heart attack. Then he reported the percentage of heart attacks for the patients who took aspirin every day and for those who did not take aspirin every day.
- Researcher 2 also studied 500 people. He randomly assigned half of the patients to take aspirin every day and the other half to take a placebo everyday. After a certain length of time, he reported the percentage of heart attacks for the patients who took aspirin every day and for those who did not take aspirin every day. Suppose that both researchers found that there is a statistically significant difference in the heart attack rates for the aspirin users and the non-aspirin users and that aspirin users had a lower rate of heart attacks.
- What is the design of each study? Can researcher 1 conclude that aspirin caused the reduction in rate of heart attacks? Why or why not?
- Source: SOCR Problem 2
- Hospital floors are usually covered by bare tiles. Carpets would cut down on noise but might be more likely to harbor germs. To study this possibility, investigators randomly assigned 8 of 16 available hospital rooms to have carpet installed. The others were left bare. Later, air from each room was pumped over a dish of agar. The dish was incubated for a fixed period, and the number of bacteria colonies were counted.
- What is the appropriate statistical term for the 8 rooms left bare. (Control group)
- What do we call the 16 rooms that were chosen for the study (sample)
- What do we call the number of colonies growing in each dish (Response variable)
- Source: SOCR Problem 4 and 7
- As I read the following article decide what you think are explan and response variables, subjects, and study design:
- Flexible Brains People who grow up left-handed have a different, more flexible brain structure than those born to take life by the right hand, says UCLA researchers who use twins to study heredity. The reason is that right-handers have genes that force their brains into a slightly more one-sided structure. Left-handers appear to be missing those genes.
- “There is a real difference in brains that result in a more symmetric brain in left-handers, where the two sides are more equal,” said UCLA neurogeneticist Dr. Daniel Geschwind, who lead the research team. “There is more flexibility and that is under genetic control.” That hereditary difference between right-handers and left-handers also appears to affect how the brain changes in size throughout a lifetime, the researchers found.
- Of all the primates, only humans display such a strong predisposition to right-handedness. Right-handers make-up about 90% of the population. The left and right halves of the brain are different in both anatomy and their features, related to hand preference. But until now, no one could document the connection.
- To study brain size and structure, the UCLA researchers used brain-scanning technique called functional magnetic resonance to compare brains in 72 pairs of identical twins, all of them male World War II veterans ages 75 to 85. Identical twins-who share the same genes- offer a unique lens through which to study the relative effects of heredity on human nature.
- Right-handers typically have a larger left brain hemisphere, where their language abilities are concentrated. Conversely, left-handers have more balances brains, with both sides relatively symmetric. “Overall, this study shows us that brain structure is highly influenced by genetics, even later in life,” Greschwind said. “This implies that aging-related changes to the brain also possess a strong genetic basis. That is kind of wild.”
- What is explanatory variable? (left and right handedness)
- What is response variable? (brain size and structure)
- Who are the subjects? (72 pairs of male twins who were WWII veterans, ages 75 to 85 years)
- What is the study design? (observational)
- Is this a matched pairs design? (not really, cause it's not an experiment; but the twins control for lurking variables to some extent)
- What can be concluded about causation?
- Source: SOCR Problems 12-15
Summaries of real research reports
Each student presents on the research report that they read:
- the question asked in the study (If there's more than one, just choose one.)
- the explanatory and response variables
- how the sample was selected (method and sampling plan)
- the study design (experimental, observational, or survey)
- the treatment groups, if any
- the outcome
- a causal diagram
This week's work
Online work: StatTutor exercise
For our next session:
- Think about possible research questions and how (design-wise) you might address the question via a research study
Design an experiment
Groups collaborate to design an experiment to address a research question. Work together to decide on the following:
- Research question
- Explanatory variable
- Values?
- Control group?
- Response variable
- How is it measured?
- Study design
- Sampling plan
Week 10 (Nov 5)
Review quiz questions
Display questions 1, 5, and 10 and discuss distractors vs. correct answer.
Review terminology in sampling and observational study design
Sampling
- What's the purpose of creating a sample? (want to know something about a population, but it's too big to study all individuals. can do study on a sample and then make inference to the population) - show The big picture. png
- What is the most important overall characteristic of a sample? (representative of population)
- What problem does a sample have if it systematically under- or over- estimates the values of a variable? (bias)
- Sampling methods (discuss how each works, examples, problems):
- volunteer sample -- (study participants volunteer to be in the study; guaranteed to be biased
- convenience sample -- (study participants chosen because right time/place for researcher; susceptible to bias because certain types of individuals are more likely to be selected than others)
- narrowly defined sample -- (study participants chosen from defined subgroup of population; subgroup may be systematically different from population)
- systematic sample -- (study participants chosen based on non-random, systematic method; predetermines who can participate, not as safe as random sampling
- simple random sample -- (study participants chosen at random from the population; volunteer response, that is, non-response, can be a problem when we can't make individuals participate)
- Including random selection in a sampling plan -- probability sampling plan (discuss how each works, examples, benefits):
- Simple random sample (SRS)
- Cluster sampling (include all members of randomly selelcted intact groups, families, classes)
- Stratified sampling (choose a SRS within groups, strata; ensures representativeness for chosen factor)
- Multi-stage sampling (stratify into successively smaller groups)
- Does sample size matter? Give an example where it does and where it doesn't.
- What is the trade-off when considering a small vs. large sample size? (generalizability and time/money)
Study design
- What is study design?
- What designs are presented in OLI Statistics? (observational study, survey, experiment)
- Ask students for example of each kind of study.
- What is an example of a prospective observational study? example of a retrospective observ. study?
Some examples for discussion
- In a large midwestern university with 30 different departments, the university is considering eliminating standardized scores from their admission requirements. The university wants to find out whether the students agree with this plan. They decide to randomly select 100 students from each department, send them a survey, and follow up with a phone call if they do not return the survey within a week. What kind of sampling plan did they use?
- (a) Stratified random sampling
- (b) Simple random sampling
- (c) Cluster sampling
- (d) Multi-stage sampling
- Source: SOCR Problem 31
- On October 20, 1993, the San Francisco Chronicle reported on a survey of top high-school students in the U.S. According to the survey: "Cheating is pervasive. Nearly 90 percent admitted some dishonesty, such as copying someones homework or cheating on an exam. The survey was sent last spring to 5,000 of the nearly 700,000 high achievers included in the 1993 edition of Who is Who Among American High School Students. The results were based on the 1,957 completed surveys that were returned.
- Is this survey representative of all teenagers? What is the population represented in this survey?
- Source: SOCR Problem 33
- In a study of Wikipedia editor behavior, the researchers randomly selected 22 of the 40 most active WikiProjects. 125 editors were then randomly selected from the group of editors participating in these 22 projects. What kind of sampling plan did they use?
- (a) Stratified random sampling
- (b) Simple random sampling
- (c) Cluster sampling
- (d) Multi-stage sampling
- Adapted from: Herding the Cats: The Influence of Groups in Coordinating Peer Production
- A radio talk show invites listeners to enter a dispute about a proposed salary increase for city council members. The host says, "What annual salary do you think council members should get? Call us with your number." In all, 958 people call. The mean of all the salaries they suggest is $9,740 per year, and the standard deviation of the responses is $1,125. Which of the following statements applies to this situation? Should the results be used to inform the debate on council members salary? Why or why not?
- Source: SOCR Problem 30
- The Democratic National Committee would like to collect the opinions of members of local democratic groups on a few issues. There are many thousands of local democratic groups, some large and some small. The researchers hired to do the study decide to focus on groups with over 1000 members, because they want the opinions of members in well-established groups. Of all of the groups with 100+ members, thirty groups are randomly selected. The survey is sent to all of the members in each of these groups. What kind of sampling plan did they use?
- (a) Stratified random sampling
- (b) Simple random sampling
- (c) Cluster sampling
- (d) Multi-stage sampling
- Suppose two researchers wanted to determine if aspirin reduced the chance of a heart attack. Researcher 1 studied the medical records of 500 patients. For each patient, he recorded whether the person took aspirin every day and if the person had ever had a heart attack. Then he reported the percentage of heart attacks for the patients who took aspirin every day and for those who did not take aspirin every day.
- Researcher 2 also studied 500 people. He randomly assigned half of the patients to take aspirin every day and the other half to take a placebo everyday. After a certain length of time, he reported the percentage of heart attacks for the patients who took aspirin every day and for those who did not take aspirin every day. Suppose that both researchers found that there is a statistically significant difference in the heart attack rates for the aspirin users and the non-aspirin users and that aspirin users had a lower rate of heart attacks.
- What is the design of each study? Can researcher 1 conclude that aspirin caused the reduction in rate of heart attacks? Why or why not?
- Source: SOCR Problem 2
This week's work
Online work:
- causation and experiments
- more than one explanatory variable
- modifications to randomization
- sample surveys
For our next session:
- Assignment is to practice identifying sampling and study design
- Ask if students want to read a real research report, using internet academic search tools.
- Students who would rather not go to the trouble of finding and reading a report, can evaluate a study summary.
Design a study
Groups collaborate to design a two-question survey of week-day morning library patrons.
- Create a sampling plan, consider the following in your plan:
- minimize bias
- at least 15 participants, at least 10% of patrons at any given time
- include an element of random selection, if possible
- Develop one-two questions to ask each patron.
- Implement your data collection plan (session takes place in room at the library)
Have the groups return by 10:15 to report on their experience.
- Identify each group's sampling method
- Identify sources of bias
- Identify sources of randomness
- Discuss effectiveness of questions asked
- Discuss problems encountered
Week 9 (Oct 29)
Questions about "Examining relationships"
- how to make box plots
- least squares
- classify two variable relationships according to case and determine appropriate displays and statistics. Some examples for discussion:
- Is there a relationship between gender and test scores on a particular standardized test? (I)
- Are the smoking habits of a person (yes/no) related to the person's gender? (II)
- How well can we predict a student's freshman year GPA from his/her SAT score? (III)
- Is there a relationship between the time a person has practiced driving while having a learner's permit, and whether or not this person passed the driving test? (IV)
- Can you predict a person's favorite type of music (Classical/Rock/Jazz) based on his/her IQ level? (IV)
- Does a the amt of sugar in a diet (lo, hi) predict activity level (normal, hyper)? (II)
- Does the amt of salt on a road predict the number of accidents? (III)
- Is the amt of fiber in food (lo, med, hi) related to the calories per serving? (I)
- StatTutor issues:
- College GPA's listed that are above 4.0. i did not know that that was possible.
- alot more people in the middle of a class rooom; how was the room split?
- did not say the size of the class.
- (display StatTutor body_image results Q2.png) OLI includes the missing cases in the total for their percent calculations. Is this generally OK? Can it cause problems? (if males and females respond at different rates)
This week's work
- Quiz (it is open note; discuss possible study strategies)
- All of module 3 on Sampling
- Begin module 4 on Designing Studies
Two-quantitative variable research questions
- Students who have changed their topic present their new research question
- Read out the research question
- Define each variable
- Describe how each variable would be measured
- Display the drawn scatterplot
- Predict whether the variables will be linearly related, and if so, describe the predicted relationship (direction, form, and strength)
Exploratory data analysis on m&m variables
Each group presents their question, results and conclusions
- Is there a case to be made that packages with large numbers of defective m&m's are generally lighter?
- What other statistics/analysis could we use to strengthen the case? (distribution of defectives (0-4), percent of packages in our sample with 2 or more defectives, scatterplot...and correlation...of defectives vs. weight)
Analyzing m&m # of defectives vs. weight and # of m&ms vs. weight
Group students into two new groups using units digit of birthday
- create a scatterplot of # of defectives predicting weight
- create a scatterplot of # of m&ms in package predicting weight
- calculate the correlation for these if appropriate
- calculate the linear regression equation for each relationship
- interpret the slope and intercept of the # of m&ms predicts weight equation (slope is the weight of 1 m&m?; intercept should be near 0 -- no m&ms means no weight). What contributes to any inconsistencies?
Week 8 (Oct 22)
Topics from linear relationships and causation
continuing with linear relationships:
- What is regression? (An analysis tool that can predict the value of the response variable given a value for the explanatory variable.)
- What is a prediction tool for two variables that are linearly related (linear regression equation)
- How is the linear regression equation calculated (least squares computation; find the line that has the smallest sum of squared vertical deviations--display Least squares concept.png)
- History of term regression: originally named by Francis Galton, cousin of Charles Darwin, to describe the phenomena that the heights of descendants of tall ancestors tend to regress down towards a normal average (a phenomenon also known as regression toward the mean)
- What is the formula for the linear regression equation? (Y = a + bX)
- (show Predicting upper level course.png) Ask students to raise hand if they answered 90 for this question (discuss use of the equation to predict Y)
- For this graph, how does the upper level course average vary with the lower level course (change of 1 on lower corresponds to change of 1 on upper)
- Only one person did the extra problems on linear relationships!
- In a 4th grade homework assignment students are given data showing the number of participants in a rec soccer league for each of the last 10 years. They are asked to predict the number of participants in the next year. What is a reasonable response? (anything is possible) Are there situations where someone might want to predict a future event?
- What is extrapolation?
- Read warning statement about predicting future results from Vanguard annual report. What are they warning people not to do? (extrapolate)
- (Display olympics.ods.) Ask how the students predicted the 1500 olympic time for 2008 (suggest using the equation -- use a calculator to determine the predicted time). What's the issue with this prediction?
Moving on to causation & lurking variables:
- What is the principle that OLI lists on every page of this section? (Association does not imply causation!)
- Examples of lurking variables in OLI. Discuss the types of variables included and name the lurking variable in each
- firefighters related to fire damage (lurking variable is seriousness of fire)
- nationality predicts SAT score (lurking variable is educational level)
- amount of light at night related to nearsightedness in children (parent's eyesight)
- death rates for particular hospitals (severity of illness)
- % taking the SAT in a state is related to median math score (prevalence of ACT or SAT in a state)
- What is Simpson's paradox? (adding a lurking variable causes us to rethink the direction of a relationship)
- Anyone have an example of Simpson's paradox.
- Other examples
- Amt of salt on the road is related to number of accidents (really obvious that relationship is not causal)
- Discuss issues with the Newsweek Back Story: "Can you cheat death". What is the article suggesting? Read some of the factors listed including "You have less than 12 years of education" and discuss which are not likely causal.
- Any other questions?
For next week
- StatTutor and My Response
- Be sure to describe the appropriate statistics and graphs when you report results.
- Use some of the most compelling data to make your conclusions.
- Look back at what you need to improve from your first StatTutor.
- Feel free to note issues you encounter in the My Response area.
- Find an instance where it's easy to assume causation based on just a correlation, develop a theory on possible lurking variable(s), and draw a causation diagram.
Exploratory data analysis on m&m variables
Each group presents their question, results and conclusions
- group work needs to represent the whole group -- the reports didn't seem like group work
- analyses: what should be included?
- Discuss what the groups put in the report:
- the question related to the m&m data that you investigated
- a data analysis section with relevant data displays (tables and charts) and a written description of the data (don't forget to include appropriate summary statistics for each variable)
- a conclusion section in which you answer the question posed using the data to support your answer
Two-quantitative variable research questions
- Students who have changed their topic present their new research question
- Read out the research question
- Define each variable
- Describe how each variable would be measured
- Display the drawn scatterplot
- Predict whether the variables will be linearly related, and if so, describe the predicted relationship (direction, form, and strength)
Week 7
Review
- What is a variable? (something that can take on different values)
- What is a statistic? (a quantity calculated from the *sample* data)
Topics from scatterplots and correlation
- When is it appropriate to graph data using a scatterplot?
- Which variable should go on the x-axis? (explanatory, if there's a clear distinction)
- When interpreting a scatterplot, what elements do we look at? (Pattern: direction, form, strength; Deviations: outliers) -- show OLI scatterplot 2 of 5
- Relating height and weight
- (display height.ods) what's the form of this relationship?
- is it appropriate to report a correlation (yes, for m/f separately)
- will the m/f correlations be similar or different (ask each for a prediction -- check that not mixing up slope with correlation)
- What is a correlation? (measure of strength and direction of linear relationship between two quant variables)
- What does a correlation look like?
- When is it inappropriate to use a correlation as a measure of strength? (nonlinear -- need to LOOK at the data)
- What are the units of measure for a correlation? (unitless)
- How do we handle changing the units for one of the variables?
- This week's work included lots of tools to help you learn to interpret a correlation. Which helped you the most?
- How can outliers effect a correlation? (single points can substantially strengthen or weaken it)
- Relating gestation period and longevity
- assign explanatory and response (actually could go either way)
- (display animals.ods) what's the effect of the elephant (strenghtens correlation)
- should we drop the elephant (outlier) from the dataset or not? If we were the researcher trying to explain this relationship what do we do? (consider getting more really big animals -- whale into the dataset)
- Any other questions?
Two-quantitative variable research questions
- Ask each student to present their research question
- Read out the research question
- Define each variable
- Describe how each variable would be measured
- Display the drawn scatterplot
- Predict whether the variables will be linearly related, and if so, describe the predicted relationship (direction, form, and strength)
Exploratory data analysis on m&m variables
- Continue group work to
- identify one question related to the m&m data collected.
- perform the necessary data analysis to answer the question.
- write up a summary including the question being addressed, relevant summary data, and conclusion.
Week 6
Topics from 'My Response' responses
- Display histogram from the quiz and discuss x-axis label issues. Discuss implication for the question "What proportion of data is less than 35?"
- Does the min/max include outliers? I can make the case to include and to exclude.
- Include outliers if the outliers are integral to the data collected;
- Exclude outliers that are considered "different" from the rest of the data (eg. error, external impact, changed conditions that affected only the aberrant data)
- For calculating things like range, when all you see is a histogram, is it best to average the bin (i.e. first bin is 0-10 and last is 50-60, would you say range is 5-55)? (from Tim) -- How does the general principle "Don't create an answer that is more precise than your data." relate to this question?
Two-variable research questions
- Review the role-type classification grid. Emphasize that unnecessary to memorize. Be sure to draw it in your notebook so you can use it to remember what's what.
- Go round robin, one question at a time,
- Read out the question
- Identify case
- Identify explanatory and response variables
- Suggest method of analysis
Understanding conditional percents
Display the tables in the AP Statistics Tutorial: Two-way tables; discuss the meaning of the cell contents in the different tables.
Select the best answer for the first two tables at statlit survey
Exploratory data analysis on m&m variables
- Continue group work to
- identify one question related to the m&m data collected.
- perform the necessary data analysis to answer the question.
- write up a summary including the question being addressed, relevant summary data, and conclusion.
Week 5
Discuss results of last week's exercises:
- standard deviation
- what is a SD?
- walk through reasoning in formula: deviation, squared deviation, divide by N-1, square root
- what is its assoc measure of central tendency? why?
- how can a SD=0?
- what are the units for the SD?
- what is a variance?
- when is it approp to use mean and SD to describe a distribution (normal, no outliers). What can be used for other types of distributions?
- Analysis of college drinking and integrity
- need to send me your spreadsheet file
- comments?
- what do you need to do to report results?
- what do you need to do to make conclusions?
- which statistics can be used to summarize the drinking histogram? (range, IQR, Q1, Q2, median)
- does the college have a drinking problem?
- Any other questions about the content before you take the quiz?
m&m weight distribution
Objective: Use the distribution of weight of fun size m&ms to inform a packaging decision.
(Recognize group that chose weight distribution for EDA project -- can continue or switch).
Organize students into the two groups created at last week's session. Have each group work through the m&m weight distribution activity ^{[1]}.
When finished with this activity, groups can continue the EDA project begun last week.
Exploratory data analysis on m&m variables
- Continue group work to
- identify one question related to the m&m data collected.
- perform the necessary data analysis to answer the question.
- write up a summary including the question being addressed, relevant summary data, and conclusion.
Week 4
List of for pre-session set-up:
- Example graduation box plots - Calc file
- Pie chart vs. bar chart
- Mercer Co. stimulus funds
- Trash bin contents
- Top 25 clipmark topics
- 3D/angled pie charts
Discuss results of last week's exercises:
- quartiles (ask for a practical example -- collegeboard.com range of middle 50% of SAT scores by measure)
- boxplot (show example boxplot, any issues?)
- 1969 Draft lottery story (discuss use of boxplots; how much do we rely on randomness to maintain fairness?)
Review uses of "average", "mean", and "median"
Have each student present his/her three examples of the use of "average", "mean", and "median" in the popular media. Discuss.
Making sense of pie charts
Discuss the following principles:
- If there are more than 6 categories, consider using a bar graph.
- Include percentage figures, and optionally the data value, for each category or slice of the pie.
- Order the slices from largest to smallest clockwise, starting at the top.
Discuss idea that humans are less able to judge the size of angles and area than heights. See this example from WP -- pie chart vs. bar chart.
Examples of problem graphs:
- Mercer Co. stimulus funds
- too many slices (ideas for a better graph?)
- funky orientation
- what is the "Multiple" category
- Trash bin contents
- too many slices (ideas for a better graph?)
- "Other" category (the other category is bigger or the same as 5 specific categories; does this undermine your visual interpretation?)
- No title (what happens when this graph is separated from the surrounding content?)
- Top 25 clipmark topics
- really, too many slices
- worse than no title, no info whatsoever
- what is the measurement?
- how much is each slice? Which slices are the same?
- what is the whole, conceptually speaking? (discuss example of group of categories that does not sum to a whole -- counting some of the trees in one block versus counting all of the shade trees in a block.)
- 3D/angled pie charts
- size perception (orange bigger than green; how does blue compare to green -- depth makes it appear bigger than it really is)
- Pie chart from Trenton Times, Feb 9 (Mercer schools seek their share in stimulus bill)
- too many slices (ideas for a better graph?)
- angled presentation undermines visual interpretation
- largest slice at a NE orientation.
Examples of misleading/erroneous/ graphs
Have each student present his/her graph and explain its problems.
Exploratory data analysis on m&m variables
- Use the formula "rand()" to assign students to one of two groups (pick a threshold beforehand)
- Have students work in groups to
- identify one question related to the m&m data collected.
- perform the necessary data analysis to answer the question.
- write up a summary including the question being addressed, relevant summary data, and conclusion.
Week 3
List of for pre-session set-up:
- Example best actor histogram - Calc file
- Oliver's fractal graphs
- two-side stemplot
Discuss results of last week's exercises:
- skewness (any trouble understanding/remembering?)
- histogram (show example histogram, discuss/demonstrate bin labeling)
- stemplot (show example two-sided stemplot).
Use of graphs to communicate statistics
Have each student present his/her identified graph. Discuss:
- Aesthetics: is the graph pleasing to look at?
- Communication: what does the graph tell us about the data? Could a table be used just as effectively?
- Other possibilities for data display: other graph choices? table?
- Possibilities for misinterpretation: what does the graph assume about the reader? Does the graph distort the data in anyway?
Debrief on m&m data collections
Discuss results of M&M fun pack variable definition and measurement and measurement process.
- What did you learn?
- Identify each variable's type.
- What questions do you have about the variables? (add them to a separate sheet in the spreadsheet file)
- Ask for ideas about what to do next (data cleaning, exploratory data analysis).
Review uses of "average", "mean", and "median"
Have each student present his/her three examples of the use of "average", "mean", and "median" in the popular media. Discuss.
Making sense of pie charts
Discuss the following principles:
- If there are more than 6 categories, consider using a bar graph.
- Include percentage figures, and optionally the data value, for each category or slice of the pie.
- Order the slices from largest to smallest clockwise, starting at the top.
Discuss idea that humans are less able to judge the size of angles and area than heights. See this example from WP -- pie chart vs. bar chart.
Examples of problem graphs:
- Mercer Co. stimulus funds
- too many slices (ideas for a better graph?)
- funky orientation
- what is the "Multiple" category
- Trash bin contents
- too many slices (ideas for a better graph?)
- "Other" category (the other category is bigger or the same as 5 specific categories; does this undermine your visual interpretation?)
- No title (what happens when this graph is separated from the surrounding content?)
- Top 25 clipmark topics
- really, too many slices
- worse than no title, no info whatsoever
- what is the measurement?
- how much is each slice? Which slices are the same?
- what is the whole, conceptually speaking? (discuss example of group of categories that does not sum to a whole -- counting some of the trees in one block versus counting all of the shade trees in a block.)
- 3D/angled pie charts
- size perception (orange bigger than green; how does blue compare to green -- depth makes it appear bigger than it really is)
Pie chart from Trenton Times, Feb 9 (Mercer schools seek their share in stimulus bill)- too many slices (ideas for a better graph?)
- angled presentation undermines visual interpretation
largest slice at a NE orientation.
Week 2
Use of statistics in everyday media
Have each student present his/her identified statistic. Discuss:
- Variable: what is being measured?
- Measurement: how is it being measured?
- Precision: is the statistic appropriately precise given it's purpose? too general? too precise?
- Ambiguity: what don't we know about this statistic?
Use of graphs to communicate statistics
Have each student present his/her identified graph.
Discuss:
- Aesthetics: is the graph pleasing to look at?
- Communication: what does the graph tell us about the data? Could a table be used just as effectively?
- Other possibilities for data display: other graph choices? table?
- Possibilities for misinterpretation: what does the graph assume about the reader? Does the graph distort the data in anyway?
Thinking about pie charts in particular
Discuss the following principles:
- If there are more than 6 categories, consider using a bar graph.
- Include percentage figures, and optionally the data value, for each category or slice of the pie.
- Order the slices from largest to smallest clockwise, starting at the top.
Discuss idea that humans are less able to judge the size of angles and area than heights. See this comparison of data presented in a pie chart vs. a bar chart.
Display the pie chart of Mercer Co. stimulus funds, what are the problems with this graph?
Are there situations in which graphing categorical data using a pie chart would be misleading? [aggregating many small categories into an "other" category; when the the categories taken together do not represent the whole, conceptually-speaking)
Share the pie chart from Trenton Times, Feb 9 (Mercer schools seek their share in stimulus bill) -- discuss its issues and how they affect interpretation. [Angled chart with too many categories, largest slice at a NE orientation, "other" category.]
Finish M&M data collection
- Students continue with M&M measurement process, until all fun packs are complete
Discuss results of M&M fun pack variable definition and measurement and measurement process.
Ask for ideas about what to do next.
Week 1
Intro stuff
Discuss the following aspects of the course:
- How's it going?
- Questions about syllabus or online schedule
- Quizzes:
- Suggest taking them after Thursday meeting, so we can review concepts if needed
- Considering 80% rule: "to move forward, a student must score 80% or higher on each quiz," or satisfactorily demonstrate knowledge of content in some other way
- Ideas for setting up Notebook/Journal
- Example:
- 2 Sep 2009: I heard an interesting statistic earlier today on the BBC morning news: "a life is lost to suicide every 30 seconds" (BBC News, 2/9/09 (14:15). I wonder how big this number is annually: 365 days x 24 hours per day x 60 min per hour x 2 half-min per min = 1,051,200 half-minutes. That's about 1 million suicides per year, let's assume globally, although the news reporter didn't say. Does this make sense?
- Estimated world population: 6.8 billion, CIA World Factbook, estimated for July 2009, accessed 2 Sept 2009.
- Estimated death rate: 8.2 deaths/1,000 population, CIA World Factbook, 2009 est., accessed 2 Sept 2009.
- Estimated current annual deaths: 55.6 million
OK, compared to 55.6 million total deaths per year, I believe that there could be 1 million of these deaths could be due to suicide. Interesting that it seems like a smaller number when compared with the total than when cited on a per 30-second basis.
Spreadsheet tips and tricks
Teacher tip
- checking your calculations (QC) -- walk through how to show color-coded formulas, using graduation.ods which has at least one error to be spotted
(possible, if time) Go around the room with each person sharing a helpful tip for using spreadsheets
- targeted find & replace
M&M data collection
Equipment needed:
- 50+ packs of fun-size M&Ms
- Scale and/or balance
- Internet access to Google spreadsheets
Procedure:
- What variables related to fun size packs of M&Ms might be interesting to study? Brainstorm a list. (Record list in text document.)
- Discuss idea of defining a variable. Use "homelessness" as an example to discuss the issues of definition.
- Define each M&M variable. (Record definition in text document.)
- Discuss idea of measuring a variable. Can all variables be measured equally well? Ask for examples.
- Define measurement method for each M&M variable. (Record measurement method in text document).
- Discuss idea of measurement process, for example census, civilian deaths in Iraq.
- Collaboratively design a process for measuring and recording the identified M&M variables. (List the steps in a text document.)
- Implement the measurement process as designed.
Other Ideas
Review of standard deviation rule
- Have students illustrate the answers on paper.
- The mean is 100 points, standard deviation is 10 points.
- Find the proportion of values below 70 points. 70 is 3 standard deviations below the mean of 100. Since 99.7% are within 3 sd's, that leaves half of 100% minus 99.7% below: .3%/2=.15%, or .0015.
- The middle 95% of the values are between what two point values? The middle 95% are within 2 standard deviations of the mean: between 80 and 120.
- The mean is 6 inches, standard deviation is 1.5 inches.
- Find the proportion of values below 6 inches. Since 6 is the mean, the proportion below it is .5.
- The shortest 16% are shorter than how many inches? Since the middle 68% are between 4.5 and 7.5, the shortest 16% are below 4.5.
Discuss ideas for shoe lab
There's all this talk about how some people (women?) have a lot of pairs of shoes. We could collect some data on this phenomenon to see for ourselves.
- What variables do we want to measure? (How should each be defined and measured.)
- Let's restrict our data collection to a particular group. What group could we use that would suit the variables we want to measure and for which we could measure the whole population.
- Decide on the design of the measurement process.
References
- ↑ Adapted from activity #2, created by A. Froelich, W. Stephenson and W. Duckworth in association with their research paper Assessment of Materials for Engaging Students in Statistical Discovery