Assumptions for the Two-Sample Independent T-Test: Pooled vs Unequal Cases

📚 Quick Study Guide

🔍 The two-sample independent t-test is used to determine if there is a statistically significant difference between the means of two independent groups. 🧪 Assumptions for Both Pooled and Unequal Variance T-Tests:
• 🌱 Independence: Observations within each sample must be independent. This means that the data points in one sample don't influence the data points in the other sample.
• 🍎 Random Sampling: Data must be collected via random sampling from each population. This ensures that the samples are representative of the populations they are drawn from.
• 📏 Interval/Ratio Scale: The dependent variable should be measured on a continuous (interval or ratio) scale.
🧪 Additional Assumptions Specific to Pooled Variance T-Test:
• 👯 Homogeneity of Variance: The variances of the two groups are equal. This can be tested using Levene's test. If this assumption is met, we 'pool' the variances to get a better estimate of the population variance.
🧪 Normality Assumption:
• 📈 Normality: Data in each group should be approximately normally distributed. This is especially important for small sample sizes. If sample sizes are large (e.g., $n > 30$), the t-test is relatively robust to departures from normality due to the Central Limit Theorem. You can check normality using histograms, Q-Q plots, or formal tests like the Shapiro-Wilk test. 🧪 Choosing Between Pooled and Unequal Variance T-Tests:
• ⚖️ If the variances are equal (homogeneity of variance is met), use the pooled variance t-test.
• 🚫 If the variances are unequal (homogeneity of variance is violated), use the unequal variance t-test (also known as Welch's t-test). Welch’s t-test does not assume equal variances.

Practice Quiz

1. What is the primary purpose of a two-sample independent t-test?
   1. A) To compare the means of two dependent groups.
   2. B) To compare the medians of two independent groups.
   3. C) To compare the means of two independent groups.
   4. D) To compare the variances of two independent groups.
2. Which of the following is an assumption of the independent samples t-test (both pooled and unequal variance)?
   1. A) The data are normally distributed.
   2. B) The observations within each sample are independent.
   3. C) The variances of the two groups are unequal.
   4. D) The sample sizes must be equal.
3. What does the assumption of 'homogeneity of variance' refer to?
   1. A) The means of the two groups are equal.
   2. B) The medians of the two groups are equal.
   3. C) The variances of the two groups are equal.
   4. D) The sample sizes of the two groups are equal.
4. Which test is typically used to assess the homogeneity of variance assumption?
   1. A) Shapiro-Wilk test
   2. B) Levene's test
   3. C) Chi-square test
   4. D) Paired t-test
5. If the homogeneity of variance assumption is violated, which version of the t-test should be used?
   1. A) Pooled variance t-test
   2. B) Dependent samples t-test
   3. C) Unequal variance t-test (Welch's t-test)
   4. D) One-sample t-test
6. What happens if the normality assumption is severely violated with small sample sizes?
   1. A) The t-test results are always valid.
   2. B) The t-test results may be unreliable.
   3. C) The homogeneity of variance assumption is automatically met.
   4. D) You should always use the pooled variance t-test.
7. For large sample sizes (e.g., n > 30), how does the Central Limit Theorem affect the normality assumption of the t-test?
   1. A) It makes the normality assumption more critical.
   2. B) It makes the t-test more robust to departures from normality.
   3. C) It requires the use of a non-parametric test.
   4. D) It only applies to the pooled variance t-test.