A manager of a company believes that the average age of its customers is 40. However, theCEO of the company thinks that the average age of the company’s customers differs from 40. To testthis, she randomly selected 60 customers, and the average customer age was found to be 42.7 years.Assume the (population) standard deviation for customer age is 8 years. Using a 5% level ofsignificance, does the sample provide enough evidence to refute the claim made by the manager
Question
A manager of a company believes that the average age of its customers is 40. However, theCEO of the company thinks that the average age of the company’s customers differs from 40. To testthis, she randomly selected 60 customers, and the average customer age was found to be 42.7 years.Assume the (population) standard deviation for customer age is 8 years. Using a 5% level ofsignificance, does the sample provide enough evidence to refute the claim made by the manager
Solution
To answer this question, we will perform a hypothesis test.
Step 1: State the null and alternative hypotheses. The null hypothesis (H0) is that the average age of the customers is 40 (μ = 40). The alternative hypothesis (H1) is that the average age of the customers is not 40 (μ ≠ 40).
Step 2: Determine the level of significance. The level of significance is given as 5% or 0.05.
Step 3: Calculate the test statistic. We will use the z-test formula since we know the population standard deviation. The formula is:
Z = (X - μ) / (σ/√n)
where: X is the sample mean = 42.7 μ is the population mean = 40 σ is the population standard deviation = 8 n is the sample size = 60
Substituting the values, we get:
Z = (42.7 - 40) / (8/√60) = 2.52
Step 4: Determine the critical value. Since this is a two-tailed test (μ ≠ 40), we look up the z-table for α/2 = 0.025. The critical value is ±1.96.
Step 5: Make a decision. Since the calculated z-value (2.52) is greater than the critical value (1.96), we reject the null hypothesis.
Conclusion: There is enough evidence at the 5% level of significance to refute the claim made by the manager that the average age of the customers is 40.
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