# Hypothesis testing - unfair deck of cards

The following demonstration may help students in their understanding of

1. the development of a null hypothesis, Ho, and an alternative hypothesis, Ha
2. collecting data as evidence against the null hypothesis
3. the need to determine the strength (p-value) of the evidence (data)

## Contents

### 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) = $\frac {10!}{0!(10-0)!}(.5)^{10} (.5)^0$ = .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.

### Resources

The following resources were used for ideas and organization in the development of this activity: