mathematics

1.60 Suppose that in Example 11.1 we wanted to determine whether there was sufficient evidence to conclude that the new system would not be costeffective. Set up the null and alternative hypotheses and discuss the consequences of Type I and Type II errors. Conduct the test. Is your conclusion the same as the one reached in Example 11.1? Explain.

·         EXAMPLE 11.1 Department Store’s New Billing System

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·         The manager of a department store is thinking about establishing a new billing system for the store’s credit customers. After a thorough financial analysis, she determines that the new system will be cost-effective only if the mean monthly account is more than $170. A random sample of 400 monthly accounts is drawn, for which the sample mean is $178. The manager knows that the accounts are approximately normally distributed with a standard deviation of $65. Can the manager conclude from this that the new system will be cost-effective?

·         SOLUTION:

·         IDENTIFY

·         This example deals with the population of the credit accounts at the store. To conclude that the system will be cost-effective requires the manager to show that the mean account for all customers is greater than $170. Consequently, we set up the alternative hypothesis to express this circumstance:

·            H1μ > 170 (Install new system)

·         If the mean is less than or equal to 170, then the system will not be cost-effective. The null hypothesis can be expressed as

·            H0μ ≤ 170 (Do not install new system)

·         However, as was discussed in Section 11-1, we will actually test μ = 170, which is how we specify the null hypothesis:

·            H0μ = 170

·            As we previously pointed out, the test statistic is the best estimator of the parameter. In Chapter 10, we used the sample mean to estimate the population mean. To conduct this test, we ask and answer the following question: Is a sample mean of 178 sufficiently greater than 170 to allow us to confidently infer that the population mean is greater than 170?

·            There are two approaches to answering this question. The first is called the rejection region method. It can be used in conjunction with the computer, but it is mandatory for those computing statistics manually. The second is the p-value approach, which in general can be employed only in conjunction with a computer and statistical software. We recommend, however, that users of statistical software be familiar with both approaches.

Exercise 12.73

·         12.73 Xr12-73 With gasoline prices increasing, drivers are more concerned with their cars’ gasoline consumption. For the past 5 years a driver has tracked the gas mileage of his car and found that the variance from fill-up to fill-up was σ2 = 23 mpg2. Now that his car is 5 years old, he would like to know whether the variability of gas mileage has changed. He recorded the gas mileage from his last eight fill-ups; these are listed here. Conduct a test at a 10% significance level to infer whether the variability has changed.

28 25 29 25 32 36 27 24

 

Exercise 12.74

12.74 Xr12-74 During annual checkups physicians routinely send their patients to medical laboratories to have various tests performed. One such test determines the cholesterol level in patients’ blood. However, not all tests are conducted in the same way. To acquire more information, a man was sent to 10 laboratories and had his cholesterol level measured in each. The results are listed here. Estimate with 95% confidence the variance of these measurements.

188 193 186 184 190 195 187 190 192 196

 

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