Statistics

Question 1. Multiple Choice – Choose the best option and then provide an explanation for why that is the best choice. (4 pts each: correct answer – 2 pts, correct explanation – 2 pts)

a) If an economist wishes to determine whether there is evidence that mean family income in a community equals $50,000
A) either a one-tail or two-tail test could be used with equivalent results.
B) a one-tail test should be utilized.
C) a two-tail test should be utilized.
D) None of the above.

b) If the p-value is less than a in a two-tail test,
A) the null hypothesis should not be rejected.
B) the null hypothesis should be rejected.
C) a one-tail test should be used.
D) no conclusion should be reached.

c) If a test of hypothesis has a Type I error probability (a) of 0.01, we mean
A) if the null hypothesis is true, we don’t reject it 1% of the time.
B) if the null hypothesis is true, we reject it 1% of the time.
C) if the null hypothesis is false, we don’t reject it 1% of the time.
D) if the null hypothesis is false, we reject it 1% of the time.

d) If the Type I error (a) for a given test is to be decreased, then for a fixed sample size n
A) the Type II error (ß) will also decrease.
B) the Type II error (ß) will increase.
C) the power of the test will increase.
D) a one-tail test must be utilized.

e) For a given sample size n, if the level of significance (a) is decreased, the power of the test
A) will increase.
B) will decrease.
C) will remain the same.
D) cannot be determined.

Question 2. Management at the Kimberly Clark Corporation need to determine how many tissues a package of Kleenex should contain. Researchers determined that 60 tissues is the mean number of tissues used during a cold. Suppose a random sample of 100 Kleenex users yielded the following data on the number of tissues used during a cold: = 52, S = 22.

a) Give the null and alternative hypotheses to determine if the number of tissues used during a cold is less than 60.

b) Using the sample information provided, calculate the value of the test statistic. Show your work.
c) Suppose the alternative we wanted to test was H1 : µ < 60. State the correct rejection region for a = 0.05.
d) Suppose the test statistic does fall in the rejection region at a = 0.05. Which of the following conclusions is correct? Justify your choice.
A) At a = 0.05, there is not sufficient evidence to conclude that the average number of tissues used during a cold is 60 tissues.
B) At a = 0.05, there is sufficient evidence to conclude that the average number of tissues used during a cold is 60 tissues.
C) At a = 0.05, there is not sufficient evidence to conclude that the average number of tissues used during a cold is not 60 tissues.
D) At a = 0.10, there is sufficient evidence to conclude that the average number of tissues used during a cold is not 60 tissues.
Question 3. We have created a 95% confidence interval for µ with the result (10, 15).

a) What decision will we make if we test H0 : µ = 16 versus H1 : µ ? 16 at a = 0.05? Justify your answer.
A) Reject H0 in favor of H1.
B) Accept H0 in favor of H1.
C) Fail to reject H0 in favor of H1.
D) We cannot tell what our decision will be from the information given.

b) What decision will we make if we test H0 : µ = 16 versus H1 : µ ? 16 at a = 0.10? Justify your answer.
A) Reject H0 in favor of H1.
B) Accept H0 in favor of H1.
C) Fail to reject H0 in favor of H1.
D) We cannot tell what our decision will be from the information given.

c) What decision will we make if we test H0 : µ = 16 versus H1 : µ ? 16 at a = 0.025?
A) Reject H0 in favor of H1.
B) Accept H0 in favor of H1.
C) Fail to reject H0 in favor of H1.
D) We cannot tell what our decision will be from the information given.
Question 4. The owner of a local nightclub has recently surveyed a random sample of n = 250 customers of the club. She would now like to determine whether or not the mean age of her customers is over 30. If so, she plans to alter the entertainment to appeal to an older crowd. If not, no entertainment changes will be made.

a) State the appropriate null hypothesis and alternative hypothesis. Explain how you obtain your answer.

b) If she wants to be 99% confident in her decision, what rejection region should she use? Explain how you obtain your answer.

c) Suppose she found that the sample mean was 30.45 years and the sample standard deviation was 5 years. If she wants to be 99% confident in her decision, what decision should she make? Explain how you obtain your answer.
d) Suppose she found that the sample mean was 30.45 years and the sample standard deviation was 5 years. If she wants to be 99% confident in her decision, what conclusion can she make? Explain how you obtain your answer.
e) Suppose she found that the sample mean was 30.45 years and the sample standard deviation was 5 years. What is the p-value associated with the test statistic? Explain how you obtain your answer.
Question 5. A major DVD rental chain is considering opening a new store in an area that currently does not have any such stores. The chain will open if there is evidence that more than 5,000 of the 20,000 households in the area are equipped with DVD players. It conducts a telephone poll of 300 randomly selected households in the area and finds that 96 have DVD players.

a) State the null hypothesis and the alternative hypothesis.
b) What is the value of the test statistic in this problem? Explain how you obtain your answer.
c) What is the p-value associated with the test statistic in this problem? Explain how you obtain your answer.

d) Using a 3% level of significance, what would be your decision in the hypothesis test? Justify your answer.
e) Should the rental chain’s open a new store based on the hypothesis test using a 3% level of significance? Justify your answer.

Question 6. You are given the following information:
s12 = 4 s22 = 6
n1 = 16 n2 = 25

a) Calculate the degrees of freedom that should be used in the pooled-variance t test. Show how you obtain your answer.
b) Calculate sp2, the pooled sample variance that should be used in the pooled-variance t test. Show how you obtain your answer.
Question 7. A researcher randomly sampled 30 graduates of an MBA program and recorded data concerning their starting salaries. Of primary interest to the researcher was the effect of gender on starting salaries. Analysis of the mean salaries of the females and males in the sample is given below. Suppose the researcher was attempting to show statistically that the female MBA graduates have a significantly lower mean starting salary than the male MBA graduates.
Hypothesized Difference 0
Level of Significance 0.05
Population 1 Sample
Sample Size 18
Sample Mean 48266.7
Sample Standard Deviation 13577.63
Population 2 Sample
Sample Size 12
Sample Mean 55000
Sample Standard Deviation 11741.29
Difference in Sample Means -6733.3
t-Test Statistic -1.40193
Lower-Tail Test
Lower Critical Value -1.70113
p-Value 0.085962

a) According to the test run, state an appropriate alternative hypothesis.

b) From the analysis in Table 10-2, what is the correct test statistic? Explain how you obtain your answer.

c) What is the 95% confidence interval estimate for the difference between two means? Show how you obtain your answer.

d) What conclusion do you reach for this test? Explain how you obtain your answer.

Question 8. Determine whether each statement is “true” or “false” and explain why.

a) “A statistics professor wanted to test whether the grades on a statistics test were the same for upper and lower classmen. The professor took a random sample of size 10 from each, conducted a test and found out that the variances were equal. For this situation, the professor should use a t test with related samples.”

b) “A statistics professor wanted to test whether the grades on a statistics test were the same for upper and lower classmen. The professor took a random sample of size 10 from each, conducted a test and found out that the variances were equal. For this situation, the professor should use a t test with independent samples.”

c) “Marine drill instructor recorded the time in which each of 11 recruits completed an obstacle course both before and after basic training. To test whether any improvement occurred, the instructor would use a t-distribution with 11 degrees of freedom.”

d) “A Marine drill instructor recorded the time in which each of 11 recruits completed an obstacle course both before and after basic training. To test whether any improvement occurred, the instructor would use a t-distribution with 10 degrees of freedom.”

Question 9. A corporation randomly selects 150 salespeople and finds that 66% who have never taken a self-improvement course would like such a course. The firm did a similar study 10 years ago in which 60% of a random sample of 160 salespeople wanted a self-improvement course. The groups are assumed to be independent random samples. Let p1 and p2 represent the true proportion of workers who would like to attend a self-improvement course in the recent study and the past study, respectively.

a) If the firm wanted to test whether this proportion has changed from the previous study, which represents the relevant hypotheses?
A) H0 : p1 – p2 = 0 versus H1: p1 – p2 ? 0
B) H0 : p1 – p2 ? 0 versus H1 : p1 – p2 = 0
C) H0 : p1 – p2 = 0 versus H1 : p1 – p2 > 0
D) H0 : p1 – p2 = 0 versus H1 : p1 – p2 < 0

b) What is the unbiased point estimate for the difference between the two population proportions? Explain how you obtain your answer.

c) What is/are the critical value(s) when performing a Z test on whether population proportions are different if a = 0.05? Explain how you obtain your answer.
d) What is the estimated standard error of the difference between the two sample proportions? Show how you obtain your answer.

e) Construct a 95% confidence interval estimate of the difference in proportion of workers who would like to attend a self-improvement course in the recent study and the past study. Show how you obtain your answer.
f) The company tests to determine at the 0.05 level whether the population proportion has changed from the previous study. Which would be the conclusion? Explain how you obtain your answer.
Question 10. The amount of time required to reach a customer service representative has a huge impact on customer satisfaction. Below is the Excel output from a study to see whether there is evidence of a difference in the mean amounts of time required to reach a customer service representative between two hotels. Assume that the population variances in the amount of time for the two hotels are not equal.

t-Test: Two-Sample Assuming Unequal Variances
Hotel 1 Hotel 2
Mean 2.214 2.0115
Variance 2.951657 3.57855
Observations 20 20
Hypothesized Mean Difference 0
df 38
t Stat 0.354386
P(T<=t) one-tail 0.362504
t Critical one-tail 1.685953
P(T<=t) two-tail 0.725009
t Critical two-tail 2.024394
a) State the null and alternative hypotheses for testing if there is evidence of a difference in the variabilities of the amount of time required to reach a customer service representative between the two hotels.

b) What is the value of the test statistic for testing if there is evidence of a difference in the variabilities of the amount of time required to reach a customer service representative between the two hotels? Explain how you obtain your answer.

c) What is the critical value for testing if there is evidence of a difference in the variabilities of the amount of time required to reach a customer service representative between the two hotels at the 5% level of significance? Explain how you obtain your answer.

d) Suppose a = 0.05. Which of the following represents the correct conclusion for a test on a difference in the variabilities of the amount of time required to reach a customer service representative between the two hotels? Show how you obtain your answer.
A) There is no evidence of a difference in the variabilities of the amount of time required to reach a customer service representative between the two hotels.
B) There is evidence of a difference in the variabilities of the amount of time required to reach a customer service representative between the two hotels.
C) There is no evidence that the variabilities of the amount of time required to reach a customer service representative between the two hotels are the same.
D) There is evidence that the variabilities of the amount of time required to reach a customer service representative between the two hotels are the same.
PLACE THIS ORDER OR A SIMILAR ORDER WITH US TODAY AND GET A GOOD DISCOUNT