Chapter 20: Testing Hypotheses About Proportions

1. Chapter 20: Testing Hypotheses About Proportions (Halfway through Unit 5 in AP Statistics)

2. Hypothesis Proving? bad word • Hypothesis = Conjecture we assume is true • Data consistent w H Fail to Reject H (support, retain) • Data inconsistent w H  Reject H What is consistent enough? Use Probability (P-value)! P-value high  Fail to reject H P-value low  Reject H Careful! We are still inferring. We can not be certain.

3. Hypotheses TypesNote: Hs get written BEFORE the study is performed! No design bias! • Null Hypothesis = H0 • Assume this is true when you begin study • “Ordinary state of affairs” • Specify parameter and its value • H0 : parameter = hypothesized value • Helps us define a null model to use for evaluating data • Alternative Hypothesis = Ha • “Unusual” state of affairs – what you’d like to show is true • Same parameter, diff value (often a range) • Ha : parameter < hypothesized value

4. Z-tests (t-tests later) • Sampling distribution models for proportions  P-values • Using this Normal model (check your conditions!), when we calc P, we call this a one-proportion z-test. • Normal model for proportions requires conditions • 1. • 2. • 3. • Z-test requires conditions from one-proportion z-interval. • 1. • 2. • 3. • 4. How are these lists different? Do I need to check for 7 things? NO!

5. Example • Claim: SproutIt improves germination. Know that Nwater(0.67, 0.047). • H0 : pSproutIt = 0.67 • Ha : pSproutIt > 0.67 • Note: This is a one-sided alternative; we are interested in the P of data being on one side of the null model. • Study: 100 seeds with SproutIt applied in water per box directions (in the same way prior studies with water were conducted). Completely random study. 70 seeds sprout. Assuming H0, what is P of seeing this proportion sprout? What is the P-value? How do we feel about rejecting or failing to reject Hs?

6. Cut-Offs • Are results plausible given H0? • Some error is inevitable • Natural variation  should want to talk about SD • Chance • How unlikely or likely? • No set cut-off. It should be based on prior information about context in question.

7. More practice 1. A research team wants to know if aspirin helps thin blood. H0 says it doesn’t. They test 12 patients, observe the proportion with thinner blood, and get a P-value of 0.32. They proclaim that aspirin doesn’t work. What do you say? (Hint: consider the null model and your assumptions) 2. An allergy drug has been tested and found to give relief to 75% of the patients in a large clinical trial. Now the scientists want to see if the new improved version works even better. What would the H0 be? 3. The new drug is tested and the P-value is 0.0001. What would you conclude about the new drug?

8. Practice Answers 1. Note that n stinks – if aspirin doesn’t help, p = 0.5 and pn = 6, qn = 6. Can’t make a null model that follows 3 required conditions. Unable to reject null hypothesis. Even if study was designed with an adequate n, 32% of the time, there is some kind of blood change (unclear metrics and model), so we could advise the research team to use sound design principles to try again. 2. H0: proportion of patients experiencing relief =75% 3. Strong evidence against H0. Reject H0 and conclude that new drug gives relief to more than 75% of patients.

9. Mashed potatoes or reconstituted potato flakes? (Taste test) • H0: proportion people liking reconst. Potato flakes = 0.5 • Ha: p ≠ 0.5 Note: This is a 2-sided alternative; p > 0.5 and p < 0.5 are included. Use 2P of one side. In a completely random study, 100 people tasted both and indicated what they preferred. • Name the null model. Proportion of people liking reconst. potato flakes was found to be 0.4. • Find P-value and write a conclusion. • Add a 95% CI for this study and state it in a sentence.

10. TI Tips STAT TESTS 5: 1-Prop Ztest P0 enter H0’s proportion x enter observed number (not a proportion) n enter sample size use arrow keys to select correct Ha Try it for the prior study! What is x? What is p-value?