STAT2507 Assignment 4 Complete Solution

Question: #1410

STAT 2507 Assignment # 4 (Chapters 9 & 10) Winter 2015
Due in class: Sections E, F and H, April 1 ; Section G, April 2

Last Name ——————————————- First Name ———————-
Student # ——————————————- Lab session:————–
Total of marks=100.

Part I. Lab questions. Use only the blanks left to answer lab questions.
1. Confidence interval for μ when ! is known Suppose n = 9 people are selected at random from a large population. Assume the heights of the people in this population are normal, with mean μ = 68.71 inches and ! = 3 inches. Simulate the results of this selection 20 times and in each case find a 90% confidence interval for μ. The following commands may be used:
MTB > random 9 c1-c20;
SUBC> normal 68.71 3.
MTB > zinterval 0.90 3 c1-c20
a. [2] How many of these intervals do you expect to include μ = 68.71?
b. [2] How many of your intervals contain μ?
c. [2] Do all the intervals have the same width? Why (what is the theoretical width)?
d. [2] Suppose you constructed 85% intervals instead of 90%. Would they be narrower or wider?
e. [2] How many of your intervals contained the value 71?
f. [1] Suppose you took samples of size n = 4 instead of n = 9. Would you expect more or fewer intervals to contain the value 71? ————– [1] What about 68.71? ————–
[1] What about the width of the intervals for n = 4?
[1] Would the width of the intervals with n = 4 be narrower or wider than with n = 9?

2. Confidence interval for μ when ! is NOT known
Repeat the simulation of Question 1 but now assume ! is unknown and use the t-intervals
command to get the 20 90% intervals:
MTB > random 9 c1-c20;
SUBC > normal 68.71 3.
MTB > tinterval 0.90 c1-c20
a. [2] How many of your intervals contain μ?
b. [2] Would you expect all 20 of the intervals to contain μ? Why?
c. [1] Do all the intervals have the same width? [2] Why (what is the theoretical width)?
d. [2] Suppose you took 95% intervals instead of 90%. Would they be narrower or wider?
e. [2] How many of your intervals contain the value 71?
f. [2] Suppose you took samples of size n = 64 instead of n = 9. Would you expect more or
fewer intervals to contain 71? ———– [2] What about 68.71? ————- [1] What about the
width of the intervals for n = 64? ————— [1] Would they be narrower or wider than for
n = 9?

3. Hypothesis testing for μ when ! is known Imagine choosing n = 16 women at random from a large population and measuring their heights. Assume that the heights of the women in this population are normal with μ = 63.8 inches and ! = 3 inches. Suppose you then test the null hypothesis H0 : μ = 63.8 versus the alternative that Ha : μ 6= 63.8, using ↵ = 0.10. Assume ! is known. Simulate the results of
doing this test 30 times as follows:

MTB > random 16 c1-c30;
SUBC > normal 63.8 3.
MTB > ztest 63.8 3 c1-c30
a. [2] In how many tests did you reject H0. That is, how many times did you make an “incorrect decision”?
b. [1] Are the p-values all the same for the 30 tests?
c. [1] Suppose you used ↵ = 0.001 instead of ↵ = 0.10. Does this change any of your decisions to reject or not? [2] In general, should the number of rejections increase or decrease
if ↵ = 0.001 is used instead of ↵ = 0.10?
d. [2] Now assume that the population really has a mean of μ = 63, instead of 63.8, and carry out the above 30 simulations, (thus, use the above minitab commands with ’normal
63.8 3’ changed to ’normal 63 3’. Once again, using ↵ = 0.10 and assuming ! known, in how many tests did you reject H0?
[1] A rejection of H0 : μ = 63.8 in part (a) is a “correct decision”. True or False?
[1] A rejection of H0 : μ = 63.8 in part (d) is a “correct decision”. True or False?

4. Hypothesis testing for μ when ! is NOT known
Repeat Question 3, using ttest instead of ztest, and answer parts (a), (b), and (c) again. (Thus
‘ztest 63.8 3 c1-c30’ changes to ‘ttest 63.8 c1-c30’)
a. [2] In how many tests did you reject H0. That is, how many times did you make an
“incorrect decision”?
b. [1] Are the p-values all the same for the 30 tests?
c. [1] Suppose you used ↵ = 0.00008 instead of ↵ = 0.10. Does this change any of your
decisions to reject or not?
d. [1] In general, should the number of rejections increase or decrease if ↵ = 0.00008 is used
instead of ↵ = 0.10?
Part II Comprehension questions

1. A fast food franchiser is considering building a restaurant at a certain location. According to
a financial analysis, a site is acceptable only if the number of pedestrians passing the location
averages more than 100 per hour. A random sample of 50 hours produced ¯x = 110 and s = 12
pedestrians per hour.

(a) [5] Do these data provide sufficient evidence to establish that the site is acceptable? Use ↵ = 0.05.
(b) [4] What are the consequences of Type I and Type II errors? Which error is more expensive to make?
(c) [2] Considering your answer in part (b), should you select ↵ to be large or small? Explain.
(d) [1] What assumptions about the number of pedestrians passing the location in an hour are necessary for your hypothesis test to be valid?

2. An experiment was conducted to test the e↵ect of a new drug on a viral infection. The infection was induced in 100 mice, and the mice were randomly split into two groups of 50. The first group, the control group, received no treatment for the infection. The second group received the drug. After a 30-day period, the proportions of survivors, ˆp1 and ˆp2, in the two groups were found to be 0.36 and 0.60, respectively.
(a) [5] Is there sufficient evidence to indicate that the drug is e↵ective in treating the viral
infection? Test at 5% significance level. (Make sure to state your null and alternative hypotheses.)
(b) [5] Use a 95% confidence interval to estimate the actual di↵erence in the cure rates, i.e.
p1 − p2, for the treatment versus the control groups. Based on this confidence interval
can you conclude that the drug is e↵ective? Why?

3. In an investigation of pregnancy-induced hypertension, one group of women with this disorder was treated with low-dose aspirin, and a second group was given a placebo. A sample consisting of 23 women who received aspirin has mean arterial blood pressure 111 mm Hg
and standard deviation 8 mm Hg; a sample of 24 women who were given the placebo has mean blood pressure 109 mm Hg and standard deviation 8 mm Hg.
(a) [5] At the 0.01 level of significance, test the null hypothesis that the two populations of women have the same mean arterial blood pressure. Justify any approach you use.
(b) [5] Construct a 99% confidence interval for the true di↵erence in population means.Does this interval contain the value 0? Based on this confidence interval, what is you conclusion regarding the e↵ect of the two treatments on the blood pressure of pregnant
women?

4. In an attempt to compare the starting salaries for university graduates who majored in education and the social sciences, random samples of 100 recent university graduates were selected from each major and the following sample information was obtained: Major Mean St. Dev.
Education $50,554 $2225
Social Science $48,348 $2375
Conduct an appropriate hypothesis test at the 5% level of significance to determine if there
is a di↵erence in the average starting salaries for all university graduates who majored in
education and the social sciences. Conduct this test using
(a) [5] the p-value method,
(b) [5] the critical value method, and
(c) [5] the confidence interval method .

5. [7] A company is interested in o↵ering its employees one of two employee benefit packages. A random sample of the company’s employees is collected, and each person in the sample is asked to rate each of the two packages on an overall preference scale of 0 to 100. Results were
Employee Program A Program B
1 45 56
2 67 70
3 63 60
4 59 45
5 77 85
6 69 79
7 45 50
8 39 46
9 52 50
10 58 60
11 70 82
At significant level ↵ = 0.05, do you believe that the employees of this company prefer, on the average, one package over the other? State the Null and Alternative hypotheses and show the calculations that you use to draw a conclusion.