How much evidence is there that the new system has reduced the mean waiting time to below six minutes?
THE BANK CUSTOMER WAITING TIME CASE WaitTime
Letting μ be the mean waiting time under the new system, we found in Exercise 9.9 that we should test H0: μ ≥ 6 versus Ha: μ < 6 in order to attempt to provide evidence that μ is less than six minutes. The random sample of 100 waiting times yields a sample mean of minutes. Moreover, Figure 9.6 gives the MINITAB output obtained when we use the waiting time data to test H0: μ = 6 versus Ha: μ < 6. On this output the label “SE Mean,” which stands for “the standard error of the mean,” denotes the quantity
, and the label “Z” denotes the calculated test statistic. Assuming that σ equals 2.47:
a Use critical values to test H0 versus Ha at each of α = .10, .05, .01, and .001.
b Calculate the p-value and verify that it equals .014, as shown on the MINITAB output. Use the p-value to test H0 versus Ha at each of α = .10, .05, .01, and .001.
c How much evidence is there that the new system has reduced the mean waiting time to below six minutes?
Figure 9.6: MINITAB Output of the Test of H0: μ = 6 versus Ha: μ < 6 in the Bank Customer Waiting Time Case
Note: Because the test statistic z has a denominator that uses the population standard deviation σ, MINITAB makes the user specify an assumed value for σ.
9.39 Again consider the audit delay situation of Exercise 8.11. Letting μ be the mean audit delay for all public owner-controlled companies in New Zealand, formulate the null hypothesis H0 and the alternative hypothesis Ha that would be used to attempt to provide evidence supporting the claim that μ is less than 90 days. Suppose that a random sample of 100 public owner-controlled companies in New Zealand is found to give a mean audit delay of days. Assuming that σ equals 32.83, calculate the p-value for testing H0 versus Ha and determine how much evidence there is that the mean audit delay for all public owner-controlled companies in New Zealand is less than 90 days.
9.40 Consolidated Power, a large electric power utility, has just built a modern nuclear power plant. This plant discharges waste water that is allowed to flow into the Atlantic Ocean. The Environmental Protection Agency (EPA) has ordered that the waste water may not be excessively warm so that thermal pollution of the marine environment near the plant can be avoided. Because of this order, the waste water is allowed to cool in specially constructed ponds and is then released into the ocean. This cooling system works properly if the mean temperature of waste water discharged is 60°F or cooler. Consolidated Power is required to monitor the temperature of the waste water. A sample of 100 temperature readings will be obtained each day, and if the sample results cast a substantial amount of doubt on the hypothesis that the cooling system is working properly (the mean temperature of waste water discharged is 60°F or cooler), then the plant must be shut down and appropriate actions must be taken to correct the problem.
a Consolidated Power wishes to set up a hypothesis test so that the power plant will be shut down when the null hypothesis is rejected. Set up the null hypothesis H0 and the alternative hypothesis Ha that should be used.
b Suppose that Consolidated Power decides to use a level of significance of α = .05, and suppose a random sample of 100 temperature readings is obtained. If the sample mean of the 100 temperature readings is , test H0 versus Ha and determine whether the power plant should be shut down and the cooling system repaired. Perform the hypothesis test by using a critical value and a p-value. Assume σ = 2.
9.41 Do part (b) of Exercise 9.40 if .
9.42 Do part (b) of Exercise 9.40 if .
9.43 An automobile parts supplier owns a machine that produces a cylindrical engine part. This part is supposed to have an outside diameter of three inches. Parts with diameters that are too small or too large do not meet customer requirements and must be rejected. Lately, the company has experienced problems meeting customer requirements. The technical staff feels that the mean diameter produced by the machine is off target. In order to verify this, a special study will randomly sample 100 parts produced by the machine. The 100 sampled parts will be measured, and if the results obtained cast a substantial amount of doubt on the hypothesis that the mean diameter equals the target value of three inches, the company will assign a problem-solving team to intensively search for the causes of the problem.
a The parts supplier wishes to set up a hypothesis test so that the problem-solving team will be assigned when the null hypothesis is rejected. Set up the null and alternative hypotheses for this situation.
b A sample of 40 parts yields a sample mean diameter of inches. Assuming σ equals .016, use a critical value and a p-value to test H0 versus Ha by setting α equal to .05. Should the problem-solving team be assigned?
9.44 The Crown Bottling Company has just installed a new bottling process that will fill 16-ounce bottles of the popular Crown Classic Cola soft drink. Both overfilling and underfilling bottles are undesirable: Underfilling leads to customer complaints and overfilling costs the company considerable money. In order to verify that the filler is set up correctly, the company wishes to see whether the mean bottle fill, μ, is close to the target fill of 16 ounces. To this end, a random sample of 36 filled bottles is selected from the output of a test filler run. If the sample results cast a substantial amount of doubt on the hypothesis that the mean bottle fill is the desired 16 ounces, then the filler’s initial setup will be readjusted.
a The bottling company wants to set up a hypothesis test so that the filler will be readjusted if the null hypothesis is rejected. Set up the null and alternative hypotheses for this hypothesis test.
b Suppose that Crown Bottling Company decides to use a level of significance of α = .01, and suppose a random sample of 36 bottle fills is obtained from a test run of the filler. For each of the following three sample means, determine whether the filler’s initial setup should be readjusted. In each case, use a critical value and a p-value, and assume that σ equals .1.
9.45 Use the first sample mean in Exercise 9.44 and a confidence interval to perform the hypothesis test by setting α equal to .05. What considerations would help you to decide whether the result has practical importance?