An article in the Son Jose Meraury News stated that students in the Califarnia state university system take 5 years, on average, to finith their undengraduate degrees. A freshman student believes that the mean time is lers and conducts a survey of 37 students. The student obtains a sample mean of 4.7 with a sample standard deviation of 1.4 . Is there sufficient evidence to support the student's claim at an \( \alpha=0.1 \) significance level? Preliminary: a. Is it safe to assume that \( \mathbf{n} \leq 5 \% \) of all college students in the local arear Yes No b. \( 15 \pi \geq 30 \) ? Yes No Test the claim: a. Determine the null and altemative hypotheses. Enter correct symbal and value. \[ \begin{array}{l} H_{0}: \mu=\square \\ H_{a}: \mu ? \mathbf{Y} \square \end{array} \] b. Determine the test statistic. Round to four decimal places. \[ t=\square \] c. Find the \( p \)-value. Round to 4 decimats. \( p \)-value \( = \) \( \square \) 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 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 years.

To determine if the mean time to complete an undergraduate degree is indeed less than 5 years, we can use a one-sample t-test. Given the sample mean of 4.7 years and a standard deviation of 1.4 based on 37 students, we have enough data to assess the student's claim against the null hypothesis that the mean equals 5 years. First off, we can validate the initial assumptions. Since the sample size of 37 is greater than 30, we can invoke the Central Limit Theorem, suggesting it's safe to analyze this sample as coming from a normally distributed population. Now, let's proceed with the hypothesis testing. 1. The null and alternative hypotheses should be expressed as: \[ \begin{array}{l} H_{0}: \mu = 5 \\ H_{a}: \mu < 5 \end{array} \] 2. For the test statistic calculation: Using the formula \( t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}} \), where \( \bar{x} = 4.7 \), \( \mu_0 = 5 \), \( s = 1.4 \), and \( n = 37 \): \[ t = \frac{4.7 - 5}{1.4/\sqrt{37}} \approx -1.8693 \] (Rounded to four decimal places, \( t \approx -1.8693 \)) 3. To find the p-value, we can use a t-distribution table or calculator with \( df = n - 1 = 36 \), resulting in: \[ p\text{-value} \approx 0.0344 \] (Rounded to four decimal places) 4. With an \( \alpha = 0.1 \), we compare the p-value to the significance level. Since \( 0.0344 < 0.1 \), we can **reject the null hypothesis**. 5. Finally, we conclude: 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 years, confirming the student’s belief impressively! 🎉 For more statistical fun, you might want to explore the world of hypothesis testing and t-tests. These concepts are essential in many fields, letting you analyze not just students’ time-to-graduate but also their performance trends over the years. The more you dive into statistics, the more powerful decisions you can make based on data! And if you want to delve deeper into real-world applications, consider how businesses use statistical tests to analyze customer satisfaction or product quality. It's fascinating to see how these principles shape decisions, from marketing strategies to quality assurance in manufacturing!