Question

A sporting goods store believes the average age of its customers is 34 or less, but a competitor wants to challenge this. A random sample of 38 customers was surveyed, and the average customer age was found to be 37.5 years. Assume the standard deviation for customer age is 7.0 years. Using a = 0.05, complete parts a through d below a. Does the sample provide enough evidence to refute the age claim made by the sporting goods store? Determine the null and alternative hypotheses. HOH HH The test statistic is (Round to two decimal places as needed.) The critical score(s) is(are) (Round to two decimal places as needed. Use a comma to separate answers as needed.) Because the test statistic the null hypothesis b. Determine the p-value for this test. The p-value is (Round to three decimal places as needed.)

Answer 1

a) Null hypothesis (Ho) : $\mu \leq 34$

Alternative hypothesis (H1) : $\mu > 34$

Test statistic is given by -

$z = \frac{\bar{x}- \mu_0}{\sigma/\sqrt{n}}$

Sample Mean = = 37.5 years

T

Pre specified value of the population mean under the null hypothesis = = 34 years

HO

Sample size = n = 38

Population standard deviation = $\sigma$ = 7 years

So, the value of the test statistic will be

$z = \frac{37.5-34}{7/\sqrt{38}}$

= 3.08

The critical value of z for one tailed test at 0.05 level of significance will be 1.65

(It can be obtained from the z table by finding the z corresponding to the area close to 0.05)

Because the test statistic > Critical value of z, we may reject the null hypothesis.

b) P value = P(z > 3.08)

= Area under the z curve on the right side of z = 3.08

= 1 - P(z < 3.08)

P(z < 3.08) can be obtained from the z table by finding the area corresponding to the z = 3.0 at 0.08 probability and is equal to 0.9990

= 1 - 0.9990

= 0.001

Since, P value < level of significance (0.05), we may reject the null hypothesis.

Hence, we conclude that there is strong evidence to refute the age claim made by sporting goods store.

So, the true population average age of the customers is greater than 34 years.

c) Since, the conclusion is to reject the null hypothesis, the researcher may have made Type I errror (which is error of rejecting the null hypothesis when it is true)

Type II errror is the error to not reject (or, accept) the null hypothesis when it is false. Since, we are already rejecting the null hypothesis, this error can't be made.

So, option (d) is the correct option.

d) The true mean customer age is 37 years and the conclusion was that the true mean age of customers is greater than 34 years, hence, they made the correct decision.

So, option (a) is the correct option.