Chapter 7: Inferences Based on a Single Sample Tests of Hypotheses
Refer to Exercise 7.99.
a. Find b for each of the following values of the population mean: 74, 72, 70, 68, and 66.
b. Plot each value of b you obtained in part a against its associated population mean. Show b on the vertical axis and m on the horizontal axis. Draw a curve through the five points on your graph.
c. Use your graph of part b to find the approximate probability that the hypothesis test will lead to a Type II error when m = 73.
d. Convert each of the b values you calculated in part a to the power of the test at the specified value of m. Plot the power on the vertical axis against m on the horizontal axis. Compare the graph of part b with the power curve of this part.
e. Examine the graphs of parts b and d. Explain what they reveal about the relationships among the distance between the true mean m and the null hypothesized mean m0, the value of b, and the power.
Manufacturers that practice sole sourcing. If a manufacturer (the vendee) buys all items of a particular type from a particular vendor, the manufacturer is practicing sole sourcing (Schonberger and Knod, Operations Management, 2001). As part of a sole-sourcing arrangement, a vendor agrees to periodically supply its vendee with sample data from its production process. The vendee uses the data to investigate whether the mean length of rods produced by the vendor's production process is truly 5.0 millimetres (mm) or more, as claimed by the vendor and desired by the vendee.
a. If the production process has a standard deviation of .01 mm, the vendor supplies n = 100 items to the vendee, and the vendee uses a = .05 in testing H0: m = 5.0 mm against Ha: m < 5.0 mm, what is the probability that the vendee's test will fail to reject the null hypothesis when in fact m = 4.9975 mm? What is the name given to this Type of error?
b. Refer to part a. What is the probability that the vendee's test will reject the null hypothesis when m = 5.0? What is the name given to this Type of error?
c. What is the power of the test to detect a departure of .0025 mm below the specified mean rod length of 5.0 mm?
Customer participation in-store loyalty card programs. Refer to the Pew Internet & American Life Project Survey (January 2016) study of 250 store customers and their participation in a store loyalty card program, Exercise 7.69 (p. 425). Recall that a store owner claimed that more than 80% of all customers would participate in a loyalty card program. You conducted a test of H0: p = .8 versus Ha: p 7 .8 using a = 01. What is the probability that the test results will support the claim if the true percentage of customers who would participate in a loyalty card program is 79%?
Specify the differences between a large-sample and a small-sample test of a hypothesis about a population mean m. Focus on the assumptions and test statistics.
Complete the following statement: The smaller the p-value associated with a test of hypothesis, the stronger the support for the _____ hypothesis. Explain your answer.
Play Golf America program. The Professional Golf Association (PGA) and Golf Digest have developed the Play Golf America program, in which teaching professionals at participating golf clubs provide a free 10-minute lesson to new customers. According to Golf Digest, golf facilities that participate in the program gain, on average, $2,400 in greens fees, lessons, or equipment expenditures. A teaching professional at a golf club believes that the average gain in greens fees, lessons, or equipment expenditures for participating golf facilities exceeds $2,400.
a. In order to support the claim made by the teaching professional, what null and alternative hypotheses should you test?
b. Suppose you selectα = 0.05. Interpret this value in the words of the problem.
c. For α = 0.05, specify the rejection region of a large sample test.
Which of the elements of a test of hypothesis can and should be specified prior to analyzing the data that are to be used to conduct the test
If you select a very small value for when conducting a hypothesis test, will tend to be big or small? Explain.
If the rejection of the null hypothesis of a particular test would cause your firm to go out of business, would you want to be small or large? Explain
A simple random sample of 25 observations was selected from a normal population. The mean and standard deviation of this sample are 20 and 5, respectively.
a. Test against at the 10% significance level.
b. Test against at the 1% significance level.