The owner of a local nightclub has recently surveyed a random sample of n = 250 customers of the club. She would now like to determine whether or not the mean age of her customers is over 30. If so, she plans to alter the entertainment to appeal to an older crowd. If not, no entertainment changes will be made. Suppose she found that the sample mean was 30.45 years and the sample standard deviation was 5 years. If she wants to be 99% confident in her decision, what conclusion can she make?
A) There is not sufficient evidence that the mean age of her customers is over 30.
B) There is sufficient evidence that the mean age of her customers is not over 30.
C) There is not sufficient evidence that the mean age of her customers is not over 30.
D) There is sufficient evidence that the mean age of her customers is over 30.

Question

Quizplus · Accepted Answer

We need to conduct a one-sample t-test to determine whether the mean age of the customers is over 30 or not.

The null hypothesis is that the mean age of the customers is equal to or less than 30:

H0: μ ≤ 30

The alternative hypothesis is that the mean age of the customers is over 30:

Ha: μ > 30

We will use a one-tailed t-test because we are interested only in the one direction of the alternative hypothesis. The level of significance is α = 0.01 because we want to be 99% confident in our decision.

The t-statistic is calculated as follows:

t = (x̄ - μ) / (s / √n)

where x̄ is the sample mean, μ is the hypothesized population mean (30), s is the sample standard deviation, and n is the sample size.

Substituting the given values, we get:

t = (30.45 - 30) / (5 / √250) = 1.27

The critical value of t for a one-tailed test with α = 0.01 and df = 249 (n-1) is 2.364.

Since the calculated t-value (1.27) is less than the critical t-value (2.364), we fail to reject the null hypothesis.

Therefore, we can conclude that there is not sufficient evidence that the mean age of the customers is over 30. The owner should not make any changes to the entertainment to appeal to an older crowd based on this sample.

The Owner of a Local Nightclub Has Recently Surveyed a Random