Question

 

The average retirement age in America is 64 years old. Do small business owners retire at a different average age? The data below shows the results of a survey of small business owners who have recently retired. Assume that the distribution of the population is normal.

57, 65, 50, 57, 65, 63, 63, 59, 54, 66, 53, 51, 68

What can be concluded at the αα = 0.01 level of significance?

1. For this study, we should use:  t-test for a population mean or z-test for a population proportion
2. The null and alternative hypotheses would be:

H0:  

H1:

3. The test statistic? t z  = _____ (please show your answer to 3 decimal places.)
4. The p-value = _______ (Please show your answer to 4 decimal places.)
5. The p-value is _____ α
6. Based on this, we should:  accept, reject, or fail to reject the null hypothesis.
7. Thus, the final conclusion is that ...
   • The data suggest the population mean is significantly different from 64 at αα = 0.01, so there is sufficient evidence to conclude that the population mean retirement age for small business owners is different from 64.
   • The data suggest the population mean retirement age for small business owners is not significantly different from 64 at αα = 0.01, so there is insufficient evidence to conclude that the population mean retirement age for small business owners is different from 64.
   • The data suggest the population mean is not significantly different from 64 at αα = 0.01, so there is sufficient evidence to conclude that the population mean retirement age for small business owners is equal to 64.

Answer 1

Solution:

First need to find mean and standard deviation from given data,

Therefore, mean and standard deviation is,

For this study, we should use: This corresponds to a two-tail test, for which a t-test for one mean, with unknown population standard deviation will be used.

The null and alternative hypotheses would be:

The number of degrees of freedom are df = n-1 = 13-1=12

Rejection Region:
Given significance level = α = 0.01 and df = 12

The p-value = 0.0169 (Please show your answer to 4 decimal places.)

The p-value is greater than α

Based on this, we should: fail to reject the null hypothesis.

Thus, the final conclusion is that The data suggest the population mean retirement age for small business owners is not significantly different from 64 at α = 0.01, so there is insufficient evidence to conclude that the population mean retirement age for small business owners is different from 64.