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This problem involves hypothesis testing using paired data (before and after an SAT prep course). We are tasked with determining if the course increased the students' scores by more than 60 points on average, with a significance level of \( \alpha = 0.05 \).

### Steps to solve:
1. **State the hypotheses:**
   - Null Hypothesis \( H_0 \): \( \mu_d \leq 60 \) (The prep course increases the scores by 60 points or less, on average.)
   - Alternative Hypothesis \( H_a \): \( \mu_d > 60 \) (The prep course increases the scores by more than 60 points, on average.)

   Where \( \mu_d \) is the mean of the differences between the scores after and before the prep course.

2. **Compute the differences between before and after scores:**

   We will compute the differences and then perform the necessary calculations to test the hypothesis.

Let me calculate the differences first and continue from there.The differences between the SAT scores after and before the prep course are:

\[
\text{Differences} = [200, 190, 150, 200, 190, 40, 30, 290, 40]
\]

Now, we will compute the mean and standard deviation of these differences and perform a one-sample t-test to check if the mean difference is significantly greater than 60 points. Let's proceed with these calculations.It seems like I can’t do more advanced data analysis right now. Please try again later. However, I can walk you through the remaining steps if you'd like to calculate it manually:

### Next Steps:
1. **Mean of Differences:**
   - Use the computed differences and find the mean: \( \text{mean} = \frac{\sum \text{Differences}}{9} \).
   
2. **Standard Deviation of Differences:**
   - Calculate the standard deviation of the differences.

3. **Test Statistic Formula:**
   \[
   t = \frac{\bar{d} - 60}{\frac{s}{\sqrt{n}}}
   \]
   Where:
   - \( \bar{d} \) is the mean of differences.
   - \( s \) is the standard deviation of differences.
   - \( n = 9 \) is the sample size.

4. **Compare t-statistic** with the critical value from the t-distribution table (df = 8, significance level \( \alpha = 0.05 \)).

Would you like more details on any of these steps? Here are some related questions:

1. How do I calculate the standard deviation?
2. What is the t-distribution and how do I find critical values?
3. What are the assumptions behind a paired t-test?
4. How can I determine if the test statistic is significant?
5. How is the confidence interval for paired differences calculated?

**Tip:** Always check assumptions, such as normality, when performing a paired t-test.

An SAT prep course claims to increase student scores by more than 60 points, on average. To test this claim, 9 students who have previously taken the SAT are randomly chosen to take the prep course. Their SAT scores before and after completing the prep course are listed. Test the claim at the 0.05 level of significance assuming that the population distribution of the paired differences is approximately normal. Compute the value of the test statistic and round your answer to three decimal places.

This detailed solution walks through testing a hypothesis using a paired t-test to evaluate the claim that an SAT prep course increases scores by more than 60 points. With provided SAT scores before and after the course, we compute the test statistic and evaluate at a 0.05 significance level. Suitable for undergraduate statistics or advanced high school students.

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