The following demonstration may help students in their understanding of
- the development of a null hypothesis, Ho, and an alternative hypothesis, Ha
- collecting data as evidence against the null hypothesis
- the need to determine the strength (p-value) of the evidence (data)
Demonstration
Estimate for completion time: 20 minutes.
Materials needed:
- Deck of cards created out of only the black cards from two matching decks
- Table of binomial cumulative distribution for n=10, p=.5
Demonstration
| An Unfair Deck of Cards
NOTE: Don't share these directions with the students.
- Present the deck of cards to the students as if it is a regular deck.
- Tell the students that you are going to select a card.
- Ask: What is the probability of selecting a red card from the deck. (The students answer 50%, 1/2 or .5...of course)
- Write: p=.5 (In fact this is the null hypothesis)
- Explain that actually you are going to select 10 cards, with replacement, and shuffling a bit in between each selection and you will count the number of red cards.
- Ask: What is the expected value? (5)
- Ask: Are we assured of getting exactly 5 red cards (no...why?...sampling variability)
- Ask: What might be a possible alternative hypothesis? (p ≠ .5...Step 1 complete...State the null and alternative hypotheses)
- Recruit two students to help: 1) to select each card and call out what color it is and 2) to tally the number of red and black.
- Run the experiment. (Be sure not to show the faces of the cards in the deck during the shuffling or otherwise.)
- The students will begin to get suspicious, maybe as early as the 3rd selection, because every card selected is black.
- Make comments such as "Wow, this is unbelievable.", "You are really lucky.", "Way to go ruining the whole experiment."
- When the experiment has concluded,
- Ask the students if they believe that p=.5 (most likely answer will be NO)
- What evidence do we have against p=.5 (the 10 card sample of all black...Step 2 complete...Collect relevant data from a random sample and summarize them)
- Using the data we have how can we make a case against p=.5, the null hypothesis?
- Ask for ideas. (hopefully someone suggests using the binomial formula to calculate probability of all black in 10 trials...P(X=0) = [math]\frac {10!}{0!(10-0)!}(.5)^{10} (.5)^0[/math] = .000976)
- Ask: How might we think about this probability? (p-value...Step 3 complete...Find the p-value, the probability of observing data like that observed assuming that Ho is true.)
- Discuss what conclusions can be made
- Ask students to draw a conclusion based on results (reject Ho...very unlikely that p=.5...Step 4 complete...Based on the p-value, decide whether or not the results are significant and draw your conclusions in context.)
- Ask: Can we say anything definitive about the deck of cards? (no...can't know what proportion of red cards are in the deck, only know it's not .5...probably less.)
- Discuss ideas for how to decide how much evidence is needed.
- Ask: At what point do you think we might have had enough evidence to reject Ho? 1/10 red? 2/10 red? 3/10 red?...
- Ask: What do we call this idea of setting the level of "what's enough evidence"? (alpha level)
- Ask: What might we use to determine our alpha level in this "experiment"? (P(X<=x)
- Display a table showing the cumulative density function for the binomial formula given N=10, p=.5. Discuss how these probabilities are based on p=.5, the null hypothesis.
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Resources
The following resources were used for ideas and organization in the development of this activity:
References
