We are given two sets of scores (before and after) for 19 students, and we are tasked with testing whether the new online learning module has improved the scores using a paired t-test. The hypotheses are:

• Null Hypothesis (H₀): μD = 0 (there is no difference in the mean scores after the new module)
• Alternative Hypothesis (H₁): μD < 0 (the new module results in lower scores)

We will conduct a paired t-test, using the following steps:

1. Calculate the difference $D$ for each pair:

$D = \text{New LM} - \text{Old LM}$

2. Calculate the sample mean $\bar{D}$ and sample standard deviation $s_D$ of the differences.

3. Use the test statistic formula for the paired t-test:

$t = \frac{\bar{D}}{s_D / \sqrt{n}}$ Where:

• $\bar{D}$ is the mean of the differences,
• $s_D$ is the standard deviation of the differences,
• $n$ is the sample size (19 in this case).

4. Calculate the degrees of freedom (df = n - 1 = 18).

5. Finally, find the p-value from the t-distribution using the calculated t-value and degrees of freedom.

Let's begin by calculating the differences, the mean, standard deviation, and then compute the test statistic and p-value.The results of the paired t-test are as follows:

• Mean of the differences ($\bar{D}$): -0.3316
• Standard deviation of the differences ($s_D$): 12.8473
• Test statistic (t): -0.1125
• p-value: 0.4558 (rounded to 4 decimal places)

Since the p-value is much larger than the significance level $\alpha = 0.05$, we fail to reject the null hypothesis. This means there is insufficient evidence to conclude that the new online learning module decreased the placement scores.

Would you like a more detailed explanation of these results or any other questions? Here are 5 additional questions for deeper understanding:

1. How does the p-value relate to the significance level?
2. What are the implications of failing to reject the null hypothesis?
3. Can we improve the sensitivity of this test by increasing the sample size?
4. How would the interpretation change if the alternative hypothesis was two-tailed?
5. What is the effect of variability in the data on the test outcome?

Tip: When conducting a hypothesis test, always verify that the assumptions of the test are met, such as normality of the population for a t-test.