Final answer:
To test the claim that the mean for all 50 year old people is less than 7.5 years, we can use a one-sample t-test with a significance level of 0.05. The test statistic is calculated using the sample mean, hypothesized mean, standard deviation, and sample size. By comparing the test statistic to the critical value from the t-distribution table, we can determine whether to reject the null hypothesis or not.
Step-by-step explanation:
To test the claim that the mean for all 50 year old people is less than 7.5 years, we can use a one-sample t-test. First, we set up the null and alternative hypotheses:
Null hypothesis (H0): The mean number of years 50 year old people expect to retire is greater than or equal to 7.5 years.
Alternative hypothesis (Ha): The mean number of years 50 year old people expect to retire is less than 7.5 years.
Next, we calculate the test statistic using the formula:
t = (sample mean - hypothesized mean) / (standard deviation / sqrt(sample size))
Using the given values, we have:
t = (7.2 - 7.5) / (3.75 / sqrt(100)) = -0.8
Finally, we compare the test statistic to the critical value from the t-distribution table at a significance level of 0.05 and degrees of freedom calculated as (sample size - 1).
If the test statistic is less than the critical value, we reject the null hypothesis in favor of the alternative hypothesis, and conclude that there is evidence to support the claim that the mean for all 50 year old people is less than 7.5 years.