THE LOGIC OF HYPHOTHESIS TESTINGClass Work for Monday, October 11

THE CANDY GAME

Want to win some candy?

Here’s how it works:Draw a card from the deck. If you get a red

card, you win candy! If you get a black card, you don’t. (Yep, that simple.)

Quick note: A standard deck of cards consists of 26 red cards (13 hearts and 13 diamonds), 26 black cards (13 spades and 13 clubs) and 2 jokers. I will remove the jokers.

QUESTIONS

1. Are you convinced the deck is not half red?

2. Why were you not convinced after the first student who played did not win candy? After the second student?

3. At what point did you become convinced?

4. You did not see many cards from the deck. Can you be absolutely positive that the deck is not half red?

PROOF OR CONVINCING EVIDENCE? It is one thing to be convinced that the deck

is not half red and another to have proof.

Our sample was all black. It would be very unlikely (though not impossible) for this to happen if the deck was really half red. In fact:

P(4 black cards in a row) = 0.06P(5 black cards in a row) = 0.03

So we became convinced when we observed something that would happen by chance only a small proportion of the time.

TESTING A HYPOTHESIZED POPULATION PARAMETER

I. Make an assumption (or observe an assumption that has been made) about a population parameter. In this case, we assumed that the proportion of red cards was 0.50.

II. Sample from the population and record the sample statistic. Our proportion of red cards was 0.00.

III. Determine how likely it would be to observe what the sample shows us if the initial assumption about the population was correct.

IV. Decide if whether or not we are convinced that the initial assumption was not correct.

MORE QUESTIONS

5. Your informal probability intuition allowed you to recognize that observing 5 losses in a row was unlikely to occur if the proportion of red cards in the deck was 0.50 (50%). This was interpreted as convincing evidence against the initial belief that the deck consisted of 50% red cards. What is mean by convincing evidence against a claim about a population proportion?

6. Suppose that 5 students had played the game and that one of them drew a red card. Would you still be convinced that the proportion of red cards was not 0.50?

7. What if 2 out of 5 students drew a red card? Would you still be convinced that the proportion of red cards was not 0.50?

LET’S MAKE IT INTERESTING…

Suppose 20 students played and only 2 drew a red card.

Or suppose 20 students played and 8 drew a red card.

When do we have convincing evidence that the deck is rigged?

RECALL FROM LAST WEEK

As the sample size increases, the sampling distribution of becomes approximately Normal. p̂

*Image from Moore’s Basic Practice of Statistics © Freeman Publishing*

USING THE SAMPLING DISTRIBUTION

8. If n = 20 and p = 0.5, draw the normal curve that represents the p-hat distribution. Label the correct value of the mean and standard deviation in this case.

9. If the sample size is 20 and the actual population proportion of successes is 0.5, would it be surprising to see a sample proportion as small as p-hat=0.4? How did you arrive at your answer?

USING THE SAMPLING DISTRIBUTION

11. If the sample size is 20 and the actual population proportion of successes is 0.5, would it be surprising to see a sample proportion as small as p-hat = 0.1? How did you arrive at your answer?

12. If the sample size is 20 and the actual population proportion of successes is 0.5, what values would you expect to see for the sample proportion? That is, what value of the sample proportion would not be surprising? Explain how you arrived at your answer.

BACK TO OUR INTERESTING QUESTIONS

13. What if 20 students played the game and only 2 drew a red card? Would you be convinced that the proportion of red cards in the deck was not 0.5?

14. What if 8 of the 20 drew a red card? Would you be convinced that the proportion of red cards in the deck was not 0.5?

15. How did the information provided by the sampling distribution of p-hat enable you to make a decision about whether there was convincing evidence against the claim that the proportion of red cards was 0.5?

EVERYONE LOVES M&MS!

16. If the claim of 40% brown (p=0.4) is correct, what would you expect to see for values of the sample proportion, p-hat?

17. What values of p-hat would you consider convincing evidence against the claim of 40% brown made by the makers of M&Ms?

The makers of M&M candies claim that 40% of plain

M&Ms are colored brown. This is a claim about a population proportion.