To determine the test statistic and p-value for this paired t-test problem, we need to follow these steps:

Step 1: Calculate the differences between pre-test and post-test values

The difference for each pair is:

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

We will compute the differences for all 20 pairs.

Step 2: Compute the mean of the differences

Let $\bar{d}$ be the mean of all differences:

$\bar{d} = \frac{\sum d_i}{n}$

Step 3: Compute the standard deviation of the differences

The standard deviation of the differences is:

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

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:

• $n$ is the sample size (number of pairs),
• $\bar{d}$ is the mean of the differences,
• $s_d$ is the standard deviation of the differences.

Step 5: Determine the p-value

Once we compute the t-statistic, we can determine the p-value using the t-distribution with $n-1$ degrees of freedom.

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