This problem is asking for a paired-sample $t$-test, as we are comparing two related sets of data (pre-test and post-test). Here's how to go about solving the problem step by step:

Step 1: State the Hypotheses

• Null Hypothesis $H_0$: $\mu_d = 0$ (there is no difference between the pre-test and post-test scores).
• Alternative Hypothesis $H_a$: $\mu_d > 0$ (the post-test scores are higher than the pre-test scores).

Step 2: Calculate the Differences

For each pair, compute the difference between the post-test and pre-test scores:

$d_i = \text{post-test} - \text{pre-test}$

Let's compute these differences:

Pre-test	Post-test	Difference $d_i$
53.6	66.4	12.8
51.8	58.9	7.1
57.7	60.4	2.7
56.2	58.6	2.4
55.4	57.5	2.1
59.1	60.8	1.7
56.4	52.1	-4.3
49.6	50.1	0.5
51.8	55.0	3.2
62.0	71.9	9.9
45.7	50.0	4.3
60.7	72.5	11.8
72.6	63.3	-9.3
46.4	53.5	7.1
46.0	51.5	5.5
50.3	53.7	3.4
72.6	66.4	-6.2
49.3	50.3	1.0

Step 3: Calculate the Mean and Standard Deviation of the Differences

Let’s denote the differences by $d_i$. Now we calculate the sample mean of the differences $\bar{d}$ and the sample standard deviation $s_d$.

$\bar{d} = \frac{1}{n} \sum_{i=1}^{n} d_i$

$s_d = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (d_i - \bar{d})^2}$

Where:

• $n$ is the number of pairs.
• $d_i$ is the difference for each pair.

Step 4: Calculate the Test Statistic

The test statistic for a paired $t$-test is given by:

$t = \frac{\bar{d}}{s_d / \sqrt{n}}$

Where:

• $\bar{d}$ is the mean of the differences.
• $s_d$ is the standard deviation of the differences.
• $n$ is the number of data pairs.

Step 5: Find the p-value

After calculating the test statistic, you can find the p-value using the t-distribution with $n-1$ degrees of freedom. This is a one-tailed test because the alternative hypothesis is $\mu_d > 0$.

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I'll calculate these steps for you now.The test statistic for the sample is $t = 2.2771$.

The corresponding p-value is $p = 0.0180$ (accurate to four decimal places).

Since the p-value $0.0180$ is greater than the significance level $\alpha = 0.001$, we do not reject the null hypothesis $H_0$. There isn't enough evidence at the 0.001 significance level to conclude that the post-test scores are significantly greater than the pre-test scores.

Would you like to ask any questions or need further clarification on the process?

Here are 5 related questions to expand the topic:

1. How does the sample size affect the outcome of a t-test?
2. What is the difference between one-tailed and two-tailed hypothesis tests?
3. Why do we use the t-distribution instead of the normal distribution in this case?
4. How can we interpret the p-value in the context of hypothesis testing?
5. What are the assumptions behind using a paired t-test?

Tip: Always check the assumptions of normality and paired structure before applying a paired-sample t-test to ensure valid results.