Question

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. A sample of 50 Escalade purchasers has average age 45 (with standard deviation 25). The manager wants to know whether this is a compelling evidence that Escalade buyers are younger on average than the typical Cadillac buyer? Use the significance level alpha = 0.05.

Answer 1

Answer:

We conclude that the Escalade buyers are younger on average than the typical Cadillac buyer.

Step-by-step explanation:

We are given that the average Cadillac buyer is older than 60, past the prime middle years that typically are associated with more spending.

A sample of 50 Escalade purchasers has average age 45 (with standard deviation 25).

Let $\mu$ = average age of an Escalade buyers.

So, Null Hypothesis, $H_0$ : $\mu$ $\geq$ 60     {means that the Escalade buyers are are of equal age or older on average than the typical Cadillac buyer}

Alternate Hypothesis, $H_A$ : $\mu$ < 60     {means that the Escalade buyers are younger on average than the typical Cadillac buyer}

The test statistics that would be used here One-sample t test statistics as we don't know about the population standard deviation;

                             T.S. =  $\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }$  ~ $t_n_-_1$

where, $\bar X$ = sample mean age of Escalade purchasers = 45

            s = sample standard deviation = 25

            n = sample of Escalade purchasers = 50

So, the test statistics  =  $\frac{45-60}{\frac{25}{\sqrt{50} } }$  ~ $t_4_9$

                                       =  -4.243

The value of t test statistics is -4.243.

Now, at 0.05 significance level the t table gives critical value of -1.677 at 49 degree of freedom for left-tailed test.

Since our test statistic is less than the critical value of t as -4.243 < -1.677, so we have sufficient evidence to reject our null hypothesis as it will fall in the rejection region due to which we reject our null hypothesis.

Therefore, we conclude that the Escalade buyers are younger on average than the typical Cadillac buyer.