A store manager claims that the average age of his customers is less than 40 years old, while the industry average is 40 years old. He asks you to test his thinking. You collect a sample of 50 customers and found that the average age is 38.7. Assuming that the population standard deviation is 12.5 years and using á = 0.05, test your manager's hypothesis. What is your conclusion?

Use a one-sample z-test.

z = (sample mean - population mean)/(standard deviation divided by the square root of the sample size)

With your data:
z = (38.7 - 40)/(12.5/√50) = ?

Finish the calculation.

Check a z-table at .05 level of significance for a one-tailed test.
If the z-test statistic exceeds the critical value from the z-table, reject the null. If the z-test statistic does not exceed the critical value from the z-table, do not reject the null.

I hope this will help get you started.

To test the manager's claim, we will use a hypothesis test. The null hypothesis (H₀) is that the average age of the manager's customers is not less than 40 years old, while the alternative hypothesis (H₁) is that the average age is less than 40 years old.

H₀: µ ≥ 40 (null hypothesis)
H₁: µ < 40 (alternative hypothesis)

We will use a one-sample t-test, as we have the sample mean, population standard deviation, and a small sample size (n=50).

To conduct the hypothesis test, we will calculate the t-statistic and compare it to the critical t-value at a significance level of α = 0.05. If the t-statistic falls in the rejection region, we will reject the null hypothesis; otherwise, we will fail to reject it.

The t-statistic can be computed using the formula:
t = (x̄ - µ₀) / (s / √n),

where x̄ is the sample mean (38.7), µ₀ is the hypothesized population mean (40), s is the population standard deviation (12.5), and n is the sample size (50).

Calculating the t-statistic:
t = (38.7 - 40) / (12.5 / √50)
t ≈ -1.38

Next, we need to find the critical t-value. Since our alternative hypothesis is one-tailed (µ < 40), we will look up the critical t-value at a significance level of α = 0.05 and degrees of freedom (df) equal to the sample size minus 1 (n-1 = 50-1 = 49).

Using a t-table or statistical software, we find that the critical t-value is -1.677 (at α = 0.05, df = 49).

Comparing the t-statistic (-1.38) to the critical t-value (-1.677), we see that the t-statistic does not fall in the rejection region. Therefore, we fail to reject the null hypothesis.

Conclusion: Based on the sample data, there is not sufficient evidence to support the claim that the average age of the store manager's customers is less than 40 years old.