Various Statistical Analyses

1. Suppose a sample of size 25 is selected from an underlying normal population, the population standard deviation is known to be 8, and the sample mean is 57. Carry out a hypothesis test at the 0.05 level of significance, where the null hypothesis H0 is  = 60 and the alternative hypothesis H1 is   60.

(a) Identify the type of hypothesis test by stating the following four characteristics: number of samples (one sample or two samples), parameter tested (mean or proportion), type of distribution used in determining the decision rule (Z or t), and number of tails (one tail or two tails).

(b) State the decision rule for the hypothesis test, using the critical value approach.

(c) Compute the test statistic. Show work.

(d) What is your statistical decision for this test? (Reject the null hypothesis or not?) Briefly explain.

(e) Determine the p-value and interpret its meaning. Using the p-value approach, what is your statistical decision? (Reject the null hypothesis or not?) Briefly explain.

2. (20 pts) In the past, the mean commuting time for all employees of Apex, Inc. has been assessed at 38 minutes. Considering the recent building boom in the region, the human resources manager suspects that the mean commuting time has increased. A random sample of 16 employees results in a sample mean commuting time of 44 minutes and a sample standard deviation of 12 minutes. To see if this sample provides good evidence that the mean commuting time has increased, the manager wishes to perform a hypothesis test at the 0.05 level of significance.

(a) Identify the type of hypothesis test by stating the following four characteristics: number of samples (one sample or two samples), parameter tested (mean or proportion), type of distribution used in determining the decision rule (Z or t), and number of tails (one tail or two tails).

(b) State the null hypothesis and the alternative hypothesis for this situation.

(c) State the decision rule for the hypothesis test, using the critical value approach.

(d) Compute the test statistic. Show work.

(e) What is your statistical decision for this test? (Reject the null hypothesis or not?) Briefly explain.

(f) Use Excel or PHStat2 to find the p-value. Using the p-value approach, what is your statistical decision? (Reject the null hypothesis or not?) Briefly explain.

(g) Based on the results of your hypothesis test, what is your conclusion about the mean commuting time for all employees at Apex, Inc.?

3. In June, 2005, Mason-Dixon Polling and Research conducted a survey of 1,100 American drivers and found that 869 of the respondents say they always wear seat belts. We would like to answer the following question: At the 0.01 level of significance, is there evidence based on the survey data that at least three fourths of American drivers say that they always wear seat belts?

(a) Identify the type of hypothesis test by stating the following four characteristics: number of samples (one sample or two samples), parameter tested (mean or proportion), type of distribution used in determining the decision rule (Z or t), and number of tails (one tail or two tails).

(b) State the null hypothesis and the alternative hypothesis for this situation:

(c) State the decision rule for the hypothesis test, using the critical value approach.

(d) Compute the test statistic. Show some work.

(e) What is your statistical decision for this test? (Reject the null hypothesis or not?) Briefly explain.

(f) Find the p-value. Using the p-value approach, what is your statistical decision? (Reject the null hypothesis or not?) Briefly explain.

(g) Based on the results of your hypothesis test, what is your answer to the original question posed in this problem?

4. A student newspaper at a college conducted a study to investigate differences between sleeping habits of male and female students. Data from two samples of students revealed the following information about the number of hours slept per night:

Male Students Female Students
nM = 43 nF = 46
hours slept hours slept
hours slept hours slept
Based on this data, can you conclude that there is any difference in the mean number of hours that students sleep each night, based on their gender, at the 0.05 level of significance?

(a) Identify the type of hypothesis test by stating the following four characteristics: number of samples (one sample or two samples), parameter tested (mean or proportion), type of distribution used in determining the decision rule (Z or t), and number of tails (one tail or two tails).

(b) Are the samples independent or matched (related)?

(c) What assumptions do you have to make about the two populations in order for the hypothesis test to be valid?

(d) State the null hypothesis and the alternative hypothesis for this situation.

(e) State the decision rule for the hypothesis test, using the critical value approach.

(f) Compute the test statistic. Show some work. (You do not need to show all of the details, but identify/state some intermediate results.)

(g) What is your statistical decision for this test? (Reject the null hypothesis or not?) Briefly explain.

(h) Use Excel or PHStat2 to find the p-value. Using the p-value approach, what is your statistical decision? (Reject the null hypothesis or not?) Briefly explain.

(i) Based on the results of your hypothesis test, what is your answer to the original question posed in this problem?

5. Ten individuals are randomly selected from the participants in a weight loss program. The individuals' weights at the beginning and at the end of the 6-week weight-loss program were compiled.

Below is a summary of the results. (More information is given than you will need!)

Individual Weight Weight Difference (= End - Beginning)
at Beginning at End Note that a negative difference means a weight loss!

1 202 197 -5
2 190 185 -5
3 177 185 8
4 160 152 -8
5 225 205 -20
6 180 184 4
7 196 185 -11
8 208 200 -8
9 185 187 2
10 177 170 -7

Variable N Mean Median StDev SE Mean
Weight, Beginning 10 190 187.5 18.535 5.862
Weight, End 10 185 185.0 15.232 4.817
Difference (End-Beginning) 10 -5.0 -6.0 8.042 2.543

(StDev is Standard Deviation, SE Mean is Standard Error of the Mean)

Based on this data, can you conclude that there is evidence that the weight loss program is successful, at the 0.05 level of significance? The meaning of success is that the mean weight difference (end weight - beginning weight) for all program participants is negative.

(a) Identify the type of hypothesis test by stating the following four characteristics: number of samples (one sample or two samples), parameter tested (mean or proportion), type of distribution used in determining the decision rule (Z or t), and number of tails (one tail or two tails).

(b) Are the samples independent or matched (related)?

(c) State the null hypothesis and the alternative hypothesis for this situation.

(d) State the decision rule for the hypothesis test, using the critical value approach.

(e) Compute the test statistic. Show some work.

(f) What is your statistical decision for this test? (Reject the null hypothesis or not?) Briefly explain.

(g) Use Excel or PHStat2 to find the p-value. Using the p-value approach, what is your statistical decision? (Reject the null hypothesis or not?) Briefly explain.

(h) Based on the results of your hypothesis test, what is your answer to the original question posed in this problem?

6. In a political poll, 462 out of 1100 randomly selected Republicans favor legislation regulating campaign financing and 279 out of 620 randomly selected Democrats favor the legislation. At the 0.05 level of significance, is there a significant difference between the proportions of Republicans and Democrats who favor the campaign financing legislation?

(a) Identify the type of hypothesis test by stating the following four characteristics: number of samples (one sample or two samples), parameter tested (mean or proportion), type of distribution used in determining the decision rule (Z or t), and number of tails (one tail or two tails).

(b) State the null hypothesis and the alternative hypothesis for this situation.

(c) State the decision rule for the hypothesis test, using the critical value approach.

(d) Compute the test statistic. Show some work. (You do not need to show all of the details, but identify/state some intermediate results.)

(e) What is your statistical decision for this test? (Reject the null hypothesis or not?) Briefly explain.

(f) Find the p-value. Using the p-value approach, what is your statistical decision? (Reject the null hypothesis or not?) Briefly explain.

(g) Based on the results of your hypothesis test, what is your answer to the original question posed in this problem?