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 greater than 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 have a level of significance at 0.01, what decision should she make?
A) Reject H₀.
B) Reject H₁.
C) Do not reject H₀.
D) We cannot tell what her decision should be from the information given.

Question

Quizplus · Accepted Answer

The null hypothesis, H₀, is that the mean age of customers is not greater than 30. The alternative hypothesis, H₁, is that the mean age is greater than 30.

At a significance level of 0.01, we need to find the critical value for a one-tailed test (since H₁ is one-tailed). Using a t-distribution with 249 degrees of freedom (since n-1 = 249), we find the critical value to be 2.33 (from a t-table or calculator).

Next, we calculate the test statistic:

t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size)) 
t = (30.45 - 30) / (5 / sqrt(250)) 
t = 1.84

Since 1.84 is not greater than 2.33, we fail to reject the null hypothesis. We do not have sufficient evidence to conclude that the mean age of customers is greater than 30, and so the nightclub owner should not alter the entertainment to appeal to an older crowd.

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