Questions
An article in the San Jose Mercury News stated that students in the California state university system take 5.5 years, on average, to finish their undergraduate degrees. A freshman student believes that the mean time is less and conducts a survey of 40 students. The student obtains a sample mean of 4.8 with a sample standard deviation of 1 . Is there sufficient evidence to support the student's claim at an $\alpha=0.01$ significance level?

Preliminary:
a. Is it safe to assume that $n \leq 5 \%$ of all college students in the local area?
No
Yes
b. Is $n \geq 30$ ?
Yes
No

Test the claim:
a. Determine the null and alternative hypotheses. Enter correct symbol and value.
\[
\begin{array}{l}
H_{0}: \mu= \\
H_{a}: \mu ? \vee
\end{array}
\]
b. Determine the test statistic. Round to four decimal places.
\[
t=
\]
c. Find the $p$-value. Round to 4 decimals.
\[
p \text {-value }=
\]
d. Make a decision.
Reject the null hypothesis.
Fail to reject the null hypothesis.
e. Write the conclusion.
There is sufficient evidence to support the claim that the mean time to complete an undergraduate degree in the California state university system is less than 5.5 years.
There is not sufficient evidence to support the claim that that the mean time to complete an undergraduate degree in the California state university system is less than 5.5 years.

Step 1 ：Identify the null and alternative hypotheses. The null hypothesis is that the mean time to complete an undergraduate degree is 5.5 years. The alternative hypothesis, which is the claim the student is trying to prove, is that the mean time is less than 5.5 years. So, \(H_{0}: \mu= 5.5\) and \(H_{a}: \mu < 5.5\).

Step 2 ：Calculate the test statistic using the formula for a t-test statistic, which is \((\text{sample mean} - \text{population mean}) / (\text{sample standard deviation} / \sqrt{\text{sample size}})\). The sample mean is 4.8, the population mean is 5.5, the sample standard deviation is 1, and the sample size is 40. The calculated test statistic is approximately \(-4.4272\).

Step 3 ：Calculate the p-value using the cumulative distribution function (CDF) for a t-distribution with degrees of freedom equal to the sample size minus 1. The CDF gives the probability of observing a value less than or equal to the given value, so to find the p-value, we need to find the CDF of the absolute value of the test statistic. The calculated p-value is approximately \(0.00003742\), which is much less than the significance level of \(0.01\).

Step 4 ：Make a decision based on the p-value. If the p-value is less than the significance level, we reject the null hypothesis. If the p-value is greater than the significance level, we fail to reject the null hypothesis. In this case, the p-value is less than the significance level, so we reject the null hypothesis.

Step 5 ：Conclude that there is sufficient evidence to support the student's claim that the mean time to complete an undergraduate degree in the California state university system is less than 5.5 years.