Homework #7, Math 320, Spring 2001

Name:____________________________      Section:____

## Math 320 Homework #7 --- Due 3/9

Include your name, section number, and homework number on every page that you hand in. Enter ``Section 1'' for the morning class (10-11AM) and ``Section 2'' for Professor Sawyer's class (12-1PM).

Begin the exposition of your work on this page. If more room is needed, continue on sheets of paper of exactly the same size (8.5 x 11 inches), lined or not as you wish, but not torn from a spiral notebook. You should do your initial work and calculations on a separate sheet of paper before you write up the results to hand in.

SHOW ALL STEPS in your calculation of confidence intervals, starting from the appropriate ``magic number'' (like 1.960 or 1.645, technically called a quantile) for a normal or Student's t-distribution. In particular, DO NOT just enter the numbers into a calculator, press a button, and write down the results. (However, you are free to check your results using a calculator. It is OK to use a calculator to find the sample mean Xbar and the sample standard deviation s.)

1. (10 points) Do exercise 6.70 on page 288. (This asks you to find a 95% confidence interval for the population standard deviation of a normal population.)

2. (10 points) Do exercise 7.6 on page 301. (This asks you to find H0 and H1 in several settings.)

3. (15 points) A lamb farmer is considering whether to add antibiotics to his lamb feed. Specifically, he or she wants to carry out a statistical test to see whether the use of antibiotics increases lamb weight.

Without antibiotics, the weights of a particular breed of lamb at one year of age are known to be normally distributed with mean mu=27.2kg and standard deviation sigma=3.7kg. The lamb farmer raises a group of n=20 lambs for one year in a special barn and measures their weights. The sample mean of the 20 lambs is Xbar=29.3kg with sample standard deviation s=3.14kg. This encourages the farmer, since he or she wants to use the antibiotics, even though the farmer has read about humans who have died from antibiotic-resistant bacteria that have been traced to the use of antibiotics in livestock feed.
(i) What is the farmer's hypothesis H0 in this case? his or her hypothesis H1? (Assume a one-sided test.)
(ii) Using T=Xbar as a test statistic, what is the P-value of the observed value of T? Assume that the sample lamb weights have the same standard deviation of sigma=3.7kg as untreated lambs, so that the normalized test statistic has a normal distribution.
(iii) Formalize the farmer's test procedure by defining a decision rule for a one-sided test with level of significance alpha=0.05 . What is the critical value for this test? What is the rejection region? Is the observed value of Xbar in the rejection region or not? Does the farmer accept or reject H0 on the basis of this decision rule?

4. (15 points) A teacher gives an exam to a class of n=16 students. The sample mean of the n=16 scores was Xbar=74.7 with sample standard deviation s=12.3  In every previous year in which the exam was given, the mean was exactly 70.0. The teacher wants to test whether this year's exam was easier than the tests of previous years (or else that this year's class was more gifted). The alternative would be that the apparently higher class average of 74.7 was just sampling error.

(i) What is the teacher's hypothesis H0? his or her hypothesis H1? (Use a one-sided test.)
(ii) Using T=Xbar as a test statistic and taking s=12.3 and n=16 into account, what is the P-value of his data? Is it less than 0.05 or greater than 0.05, using a one-sided test?
(iii) Formalize the decision procedure by defining a decision rule for a one-sided test with level of significance alpha=0.05 . What is the critical value for this test? What is the rejection region? Is the observed value of T=Xbar in the rejection region or not? Does the teacher accept or reject H0 on the basis of this decision rule?

5. (10 points) (Suggested by exercise 7.36 on page 324.) When negotiating with an insurance company, the owner of a pizza delivery service asserts that at least 75% of his drivers wear seat belts at any given time. However, in a random sample of n=8 of his drivers, only two were wearing seat belts. Is this sufficient evidence to reject the hypothesis that 75% of the drivers wear seat belts at any given time? Find the P-value using a lower one-tailed test. Find the P-value exactly from the binomial distribution, for example from Table A1.