1. The Bureau of Labor Statistics reported that the average yearly income of executives in the year 2005 was $110,000. A sample of 81 executives, which was taken in 2006, showed an average yearly income of $120,000. See Problem 21 on Excel Sheet for data.
a. We want to test to determine if there has been a significant increase in the average yearly income of executives. Develop the null and alternative hypothesis for this problem.
b. Test this problem at 99% confidence and provide conclusion.
c. Would the problem have a similar conclusion if we knew the population standard deviation was $40,000? Why?
2. Zip, Inc. manufactures Zip drives on two different manufacturing processes. Because the management of this company is interested in determining if process 1 takes less manufacturing time, they selected independent samples from each process. The results of the samples are shown below.
Process 1 Process 2
Sample Size 200 43
Sample Mean (in minutes) 10 12
Population Variance 16 25
a. State the null and alternative hypotheses
b. At 99% confidence, test to determine if there is sufficient evidence to indicate that process 2 takes a significantly longer time to manufacture the Zip drives.
c. At 99% confidence, are the two processes significantly different from one another?
3. A production supervisor at a major chemical company wishes to determine whether a new catalyst, catalyst XA-100, increases the mean hourly yield of a chemical process beyond the current mean hourly yield, which is known to be roughly equal to, but no more than, 750 pounds per hour. To test the new catalyst, 35 trial runs using catalyst XA-100 are made. The resulting yields (in pounds per hour) are found on our Excel worksheet. Assuming all factors affecting yields of the process have been held constant as possible during the test runs, it is reasonable to regard the 35 yields obtained using the new catalyst as random samples of the population.
a. Find a 95% confidence interval for the mean of all possible yields obtained using catalyst XA-100.
b. Based on this confidence interval, can we be 95% confident that the mean yield using catalyst XA-100 exceeds 750 pounds per hour? Why?
c. As a conservative measure, let’s assume that the population standard deviation is the same as the sample standard deviation. Now, determine the number of trial runs of the chemical process needed to make us 99% confident that our sample mean is within five pounds of μ.
4. In the book Essentials of Marketing Research, William R. Dillon, Thomas J. Madden, and Neil H. Firtle (1993) present pre-exposure and post-exposure attitude scores from an advertising study involving 10 respondents. The data can be found on our Excel worksheet. Assuming that the differences between pairs of post-exposure and pre-exposure scores are Normally distributed:
d. State the null and alternative hypotheses needed to attempt to establish that the advertisement increases the mean attitude score (that is, that the mean post-exposure attitude score is higher than the mean pre-exposure attitude score).
e. Test this hypothesis at α=.01 What is your conclusion?
f. Test this hypothesis at α=.001 What is your conclusion?
g. Is there a difference in your conclusions for parts b. and c.? If so, why does this difference exist?
5. The Tampa Bay (FLA) Area Chamber of Commerce wanted to know whether the mean weekly salary of nurses was larger than that of school teachers. It is believed that the historic variances of each group’s weekly salary are not the same. To investigate, they collected salary information on the amounts earned last week by a sample of school teachers and nurses. The data can be found on our Excel worksheet.
h. Is it reasonable to conclude that the mean weekly salary of school teachers is lower at a .05 level of significance? Why?
i. Are the two groups significantly different at a .05 level of significance? Explain how you arrived at this conclusion.
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