Common Mistakes When Interpreting Paired-Sample T-Test Results

📚 Quick Study Guide

• 🧮 The paired-sample t-test is used to determine if the mean difference between two sets of observations is zero. These observations are often from the same subjects before and after a treatment.
• 📊 The null hypothesis ($H_0$) states that there is no difference between the means of the two related groups. The alternative hypothesis ($H_1$) states that there *is* a difference.
• 📝 The t-statistic is calculated as: $t = \frac{\bar{d}}{s_d / \sqrt{n}}$, where $\bar{d}$ is the mean difference, $s_d$ is the standard deviation of the differences, and $n$ is the number of pairs.
• 📈 The p-value represents the probability of observing a test statistic as extreme as, or more extreme than, the one calculated if the null hypothesis is true.
• 🧐 A common significance level ($\alpha$) is 0.05. If the p-value is less than $\alpha$, we reject the null hypothesis.
• 🔑 Failing to consider the direction of the effect (positive or negative) can lead to incomplete or misleading interpretations.
• 🛑 Correlation does not equal causation. A significant paired t-test does not prove that one variable *caused* the change in the other.

🧪 Practice Quiz

1. What is the primary purpose of a paired-sample t-test?
   1. A) To compare the means of two independent groups.
   2. B) To compare the variances of two independent groups.
   3. C) To compare the means of two related groups.
   4. D) To compare the medians of two related groups.
2. Which of the following is a common null hypothesis ($H_0$) for a paired-sample t-test?
   1. A) The means of the two groups are equal.
   2. B) The mean difference between the two groups is zero.
   3. C) The variances of the two groups are equal.
   4. D) The mean difference between the two groups is non-zero.
3. What does the p-value represent in the context of a paired-sample t-test?
   1. A) The probability that the null hypothesis is true.
   2. B) The probability of observing the data if the alternative hypothesis is true.
   3. C) The probability of observing a test statistic as extreme as, or more extreme than, the one calculated if the null hypothesis is true.
   4. D) The probability of making a Type I error.
4. If the p-value is 0.03 and the significance level ($\alpha$) is 0.05, what is the correct conclusion?
   1. A) Fail to reject the null hypothesis.
   2. B) Reject the alternative hypothesis.
   3. C) Reject the null hypothesis.
   4. D) Accept the null hypothesis.
5. What is a crucial assumption to consider when interpreting paired-sample t-test results?
   1. A) The data must be normally distributed.
   2. B) The variances of the two groups must be equal.
   3. C) The differences between the paired observations must be normally distributed.
   4. D) The sample size must be greater than 30.
6. A significant paired t-test result indicates:
   1. A) A causal relationship between the two variables.
   2. B) A correlation between the two variables.
   3. C) A statistically significant difference between the means of the two related groups.
   4. D) That the null hypothesis is true.
7. What does a negative mean difference in a paired t-test suggest (assuming the first group is subtracted from the second)?
   1. A) The first group had higher scores on average.
   2. B) The second group had higher scores on average.
   3. C) There was no difference between the groups.
   4. D) The test is invalid.