Homework #6, Math 320, Spring 2001

Name:____________________________      Section:____

## Math 320 Homework #6 --- Due 3/2

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.)

1. (Somewhat like exercise 6.24 on page 267.) Ninety seven (97) men who complained of heart symptoms were told to lose weight. Of these, 51 followed a hospital-provided diet and also exercised. These individuals lost amounts X1, X2, ..., X51 with a sample mean Xbar=7.4kg and sample standard deviation s=5.4kg (n=51). The remaining 46 men exercised without dieting and lost amounts Y1, Y2, ..., Y46 with sample mean Ybar=4.6kg and sample standard deviation s=4.1kg (n=46). Find 95% confidence intervals for the true population average amount of weight loss for both groups. (Use normal-approximation confidence intervals.)

2. (Like 6.14 on page 258.) Creditors of a bank are concerned about the number of loans that the bank has granted that are now in default. An audit of 75 loans from the bank showed that 11 of the loans were in default. Find a 90% confidence interval for the proportion of loans at that bank that are in default. (You can use a normal approximation. Note that this is a 90% confidence interval, not 95%.)

3. Do exercise 6.18 on page 258.

4. The Missouri public health department is concerned with the proportion of wild skunks that are infected with a virus that is associated with human meningitis. A random sample of 14 wild skunks had no skunks (that is, 0 skunks) that were infected with this virus.

(i) Find an exact 95% confidence interval for the true population proportion of wild skunks in Missouri that are infected with this virus.
(ii) In this case, what 95% confidence interval would you obtain using the normal approximation? Which of the two confidence intervals is more reasonable? (Hint: Consider Table A6.)

5. (Like 6.32 on page 268.) A sample of 8 scores on an exam yielded the scores 85,  78,  52,  66,  74,  83,  49,  and 62. Assume that the scores are normally distributed about some unknown mean. Find a 95% confidence interval for the unknown mean. (Hint: Do not use a normal-approximation confidence interval, since the sample size is too small.)

6. (Like 5.80 on page 233.) An airline deliberately overbooks on some flights because it knows that some passengers who make reservations will not show up for a flight. Suppose that the airline sells ticket to n individuals, and that each passenger with a ticket has probability 0.90 of showing up for a particular flight. That flight can hold up to 200 passengers. What is the largest value of n so that the airline can be at least 95% certain that every person who shows up for the flight with a ticket will have a seat on the plane?

(Hint: If X is the number of ticket holders who actually show up, then E(X)=np=n(0.90). If the airline solves n(0.90)=200 and sells tickets to n=200/0.90=222.22 individuals, then the airline will be in trouble approximately 50% of the time. Thus the airline wants to sell to n people with 200 < n < 222 such that Prob(X<=200)>=0.95. Use a normal approximation.)