9.2

For each of the following, is the statement a null hypothesis, or an alternative hypothesis? Why?

a. The mean IQ of all students at Lake Wobegon High School is larger than 100.

b. The probability of rolling a 6 with a particular die equals 1/6.

c. The proportion of all new business enterprises that remain in business for at least five years is less than 0.50.

Problem 1:

For the test of H0 : p = 0.48, the z test statistic equals -1.66. The only R code

you are allowed to use is:

pnorm(-1.66, lower=F)

## [1] 0.9515428

(a) Find the p-value for Ha : p > 0.48.

(b) Find the p-value for Ha : p = 0.48

(c) Based on your answer to (a) and/or (b), infer the p-value for Ha : p < 0.48.

Explain how you used (a) and/or (b) to come up with your answer.

9.24

How to sell a burger: A fast-food chain wants to compare two ways of promoting a new burger (a turkey burger). One way uses a coupon available in the store. The other way uses a poster display outside the store. Before the promotion, their marketing research group matches 50 pairs of stores. Each pair has two stores with similar sales volume and customer demographics. The store in the pair that uses coupons is randomly chosen, and after a month-long promotion, the increases in sales of the turkey burger are compared for the two stores. The increase was higher for 28 stores using coupons and higher for 22 stores using the poster. Is this strong evidence to support the coupon approach, or could this outcome be explained by chance. Answer by performing all five steps of a two-sided significance test about the population proportion of times the sales would be higher with the coupon promotion.

Problem 2: The Michigan Democratic primary took place on March 8th of this year. For the remainder of the problem let

pH = the proportion of Michigan democrats who prefer Hillary Clinton to Bernie Sanders.

1. On March 3rd, Monmouth released a poll from a sample of 81 Michigan democrats wherein 62% preferred Hillary Clinton to Bernie Sanders (i.e 50 of 81 preferred Hillary Clinton). Based on this poll, what would be your best guess of pH?

2. Suppose, for this part of the problem, that 58% of Michigan democrats preferred Hillary Clinton to Bernie Sanders (i.e., pH = 0.58). State the sampling distribution of pH for a sample of size 81 based on this information.

3. What is the probability of Monmouth observing a sample of size 81 where 62% or more of those polled preferred Hillary Clinton if pH = 0.58 (as we assumed in the previous question)?

4. Based on your answer to the previous question, what would be your conclusion about the hypotheses

H0 :pH =0.58

Ha :pH >0.58

based on the Monmouth poll (assuming a significance level of = 0.05)?

5. On March 8th, when the primary took place, Bernie Sanders beat Hillary Clinton by 2%. That is, we learned that in truth, pH = 0.49. This led many analysts to question the validity of the pre-primary polls. Given that pH = 0.49, what is the probability of seeing a poll from a sample of size 81 where pH 0.62? Do you agree with analysts that the polling methods must be flawed? Use a statistical argument to support your claim.

R Problem: A 2006 study1 considered whether dogs could be trained to detect lung or breast cancer by the smell of a subjects breath. Five ordinary household dogs were trained to distinguish, by scent alone, exhaled breath samples of 55 lung cancer patients and 31 breast cancer patients from those of 83 control patients. Once trained, researchers tested the dogs abilities to detect cancer on a new set of samples not previously encountered by the dogs. They had each dog try to identify the one cancer sample (where a sample is exhaled breath) from a set of five samples; one of which was from a cancer patient, four of which were from controls. Let p denote the probability a dog correctly detects the cancer sample from the five samples.

1. Write out the null and alternative hypothesis corresponding to whether the dogs pre- diction is better than random guessing in the researchers test.

2. In one sample of 83 sets (where a set is the five patients, one of whom had cancer), the dogs correctly identified the cancer sample 81 times. Perform the five steps of the hypothesis test corresponding to the hypotheses you stated in the previous part. State your conclusion clearly about the dogs ability to detect cancer.

3. Perform the same hypothesis test in R using the prop.test() function.