In Exercises 5–16, use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal.


Do Men Talk Less than Women? Listed below are word counts of males and females in couple relationships (from Data Set 14 “Word Counts” in Appendix B).


a. Use a 0.05 significance level to test the claim that men talk less than women.

Verified step by step guidance

1. Step 1: Formulate the null and alternative hypotheses. The null hypothesis (H₀) states that there is no difference in word counts between men and women, or men talk as much as women. The alternative hypothesis (H₁) states that men talk less than women. Mathematically, H₀: μ₁ = μ₂ and H₁: μ₁ < μ₂, where μ₁ is the mean word count for men and μ₂ is the mean word count for women.
2. Step 2: Calculate the differences between paired samples (word counts for men and women). For each pair, subtract the word count for men from the word count for women. This will give you the differences (d). For example, for the first pair: d = 21,261 - 13,560.
3. Step 3: Compute the mean (d̄) and standard deviation (s_d) of the differences. Use the formulas for the mean and standard deviation: d̄ = (Σd) / n and s_d = sqrt((Σ(d - d̄)²) / (n - 1)), where n is the number of pairs.
4. Step 4: Perform a t-test for paired samples. Calculate the test statistic t using the formula: t = (d̄ - 0) / (s_d / sqrt(n)), where d̄ is the mean of the differences, s_d is the standard deviation of the differences, and n is the number of pairs.
5. Step 5: Compare the calculated t-value to the critical t-value from the t-distribution table at a significance level of 0.05 and degrees of freedom (df = n - 1). If the calculated t-value is less than the critical t-value, reject the null hypothesis and conclude that men talk less than women. Otherwise, fail to reject the null hypothesis.

Key Concepts

Paired Sample Data

Paired sample data involves two related groups where each member of one group is matched with a member of another group. In this context, the word counts of men and women in couple relationships are paired, allowing for a direct comparison of their communication levels. This method is essential for analyzing differences while controlling for individual variability.

Hypothesis Testing

Hypothesis testing is a statistical method used to determine if there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis. In this case, the null hypothesis states that men do not talk less than women, while the alternative hypothesis suggests that they do. The significance level of 0.05 indicates the threshold for determining statistical significance in the results.

Normal Distribution

Normal distribution is a probability distribution that is symmetric about the mean, indicating that data near the mean are more frequent in occurrence than data far from the mean. The assumption that the differences in word counts have an approximately normal distribution is crucial for applying certain statistical tests, such as the t-test, which relies on this property to validate the results.