Questions: A national business magazine reports that the mean age of retirement for women executives is 61.2. A women's rights organization believes that this value does not accurately depict the current trend in retirement. To test this, the group polled a simple random sample of 97 recently retired women executives and found that they had a mean age of retirement of 60.6. Assuming the population standard deviation is 2.8 years, is there sufficient evidence to support the organization's belief at the 0.02 level of significance?

Solution Steps

The standard error \( SE \) is calculated using the formula:

\[ SE = \frac{\sigma}{\sqrt{n}} = \frac{2.8}{\sqrt{97}} \approx 0.2843 \]

The test statistic \( Z \) is calculated using the formula:

\[ Z = \frac{\bar{x} - \mu_0}{SE} = \frac{60.6 - 61.2}{0.2843} \approx -2.1105 \]

For a two-tailed test, the P-value is calculated as:

\[ P = 2 \times (1 - T(|z|)) \approx 0.0348 \]

We compare the P-value to the significance level \( \alpha = 0.02 \):

• Since \( P \approx 0.0348 > 0.02 \), we fail to reject the null hypothesis.

We conclude that there is insufficient evidence at a \( 0.02 \) level of significance to support the belief that the mean age of retirement for women executives is not \( 61.2 \).

Final Answer