Final answer:
To test the claim that the mean time for all college students is greater than 4.5 years, a one-sample t-test can be performed. The null hypothesis is that the mean time is 4.5 years, and the alternative hypothesis is that the mean time is greater than 4.5 years. The test statistic can be calculated using the given sample mean, sample standard deviation, and significance level.
Step-by-step explanation:
To test the claim that the mean time for all college students is greater than 4.5 years, we can perform a one-sample t-test. The null hypothesis (H0) is that the mean time for all college students is 4.5 years, and the alternative hypothesis (Ha) is that the mean time is greater than 4.5 years.
Using the given information, the sample mean is 4.8 years, the sample standard deviation is 2.2 years, and the significance level is 0.05. With a sample size of 81, we can assume that the sample mean follows a normal distribution. We can calculate the test statistic using the formula:
t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))
From the calculated test statistic, we can compare it to the critical value from the t-distribution table with degrees of freedom equal to sample size minus 1.
If the calculated t-value is greater than the critical value, we reject the null hypothesis and conclude that the mean time for all college students is greater than 4.5 years. If the calculated t-value is less than or equal to the critical value, we fail to reject the null hypothesis and do not have enough evidence to support the claim that the mean time is greater than 4.5 years.