To determine the test statistic and p-value for this paired sample, we need to carry out a hypothesis test based on the paired differences between the pre-test and post-test scores for each individual. Here is the detailed process:

Step 1: Calculate the Differences

Define $D = X_1 - X_2$, where $X_1$ represents the pre-test scores and $X_2$ the post-test scores. Compute the differences for each pair.

Step 2: Calculate the Sample Mean and Sample Standard Deviation of the Differences

Once we have the differences $D$, calculate:

1. The mean of the differences, $\bar{D}$.
2. The standard deviation of the differences, $s_D$.

Step 3: Set Up the Hypotheses

Given the context, let's assume we are testing whether there is a significant difference between the pre-test and post-test scores. This is a two-tailed test.

The hypotheses can be defined as:

• Null hypothesis: $H_0: \mu_D = 0$
• Alternative hypothesis: $H_1: \mu_D \neq 0$

Step 4: Calculate the Test Statistic

The test statistic for the paired $t$-test is given by: $t = \frac{\bar{D}}{s_D / \sqrt{n}}$ where $n$ is the number of pairs.

Step 5: Find the $p$-value

Using the $t$-distribution with $n - 1$ degrees of freedom, find the two-tailed $p$-value for the computed $t$-statistic.

I'll carry out these calculations to get the precise values.The test statistic for this sample is approximately:

$\text{test statistic} = 1.8220$

The $p$-value for this sample is approximately:

$\text{p-value} = 0.0861$

If you have further questions about these calculations or need additional details, please let me know.

Additional Questions for Practice

1. What would change if the test were one-tailed instead of two-tailed?
2. How do we interpret the p-value in terms of the significance level?
3. If the pre-test and post-test were from different individuals (independent samples), how would the test change?
4. How would the conclusion differ if the p-value was less than the significance level?
5. What is the importance of checking that the differences are normally distributed?

Tip:

When conducting a paired t-test, always check if your data meet the assumption of normality for the differences, especially with small sample sizes.