Refer to the analysis of car buyers in Exercise 46.  If the population mean age for purchasers of the  Cadillac Escalade is 50 years (with  years),  then what is the probability that the test procedure used in this question will fail to reject ?

Exercise 46

Brand managers become concerned if they discover that customers are aging and gradually moving out of the high-spending age groups. For example, the average Cadillac buyer is older than 60, past the prime middle years that typically are associated with more spending. Part of the importance to Cadillac of the success of the Escalade model has been its ability to draw in younger customers. If a  sample of 50 Escalade purchasers has average   age 45 (with standard deviation 25), is this compelling evidence that Escalade buyers are younger on average than the typical Cadillac buyer?   (Assume that the data meet the sample size   condition.)

(a)  State the null and alternative hypotheses.   Describe the parameters.

(b)  Identify the Type I (false positive) and Type II  (false negative) errors.

(c)  Find the -value of the test using a normal model for the sampling distribution. Do the data supply  enough evidence to reject the null hypothesis if  ?