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).


b. Construct the confidence interval that could be used for the hypothesis test described in part (a). What feature of the confidence interval leads to the same conclusion reached in part (a)?

Verified step by step guidance

1. Step 1: Calculate the differences between the paired samples (word counts of men and women). For each pair, subtract the word count of men from the word count of women. This will give you the differences for each pair.
2. Step 2: Compute the mean of the differences. Add up all the differences obtained in Step 1 and divide by the number of pairs to find the average difference.
3. Step 3: Calculate the standard deviation of the differences. Use the formula for standard deviation: \( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \), where \( x_i \) are the differences, \( \bar{x} \) is the mean of the differences, and \( n \) is the number of pairs.
4. Step 4: Determine the margin of error for the confidence interval. Use the formula \( E = t \cdot \frac{s}{\sqrt{n}} \), where \( t \) is the critical value from the t-distribution for the desired confidence level, \( s \) is the standard deviation of the differences, and \( n \) is the number of pairs.
5. Step 5: Construct the confidence interval. Add and subtract the margin of error (calculated in Step 4) to the mean of the differences (calculated in Step 2). The resulting interval will be \( [\bar{x} - E, \bar{x} + E] \). Analyze whether the interval includes zero to determine the conclusion regarding the hypothesis test.

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. This design is often used in studies to compare two conditions or treatments, allowing for the control of variability between subjects. In this case, the word counts of men and women in couple relationships are paired, enabling a direct comparison of their communication patterns.

Confidence Interval

A confidence interval is a range of values, derived from sample statistics, that is likely to contain the true population parameter with a specified level of confidence, typically 95%. In hypothesis testing, constructing a confidence interval helps to determine if the observed difference between groups is statistically significant. If the interval does not include zero, it suggests a significant difference in the means of the two groups being compared.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. It involves formulating a null hypothesis (no effect or difference) and an alternative hypothesis (there is an effect or difference), then using sample data to determine whether to reject the null hypothesis. The results of the hypothesis test can be supported or contradicted by the confidence interval, providing a comprehensive view of the data.