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.


Heights of Presidents A popular theory is that presidential candidates have an advantage if they are taller than their main opponents. Listed are heights (cm) of presidents along with the heights of their main opponents (from Data Set 22 “Presidents” 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 heights of the presidents and their main opponents for each pair. For example, subtract the height of the opponent from the height of the president for each row in the table.
2. Step 2: Compute the mean of the differences obtained in Step 1. This will represent the average difference in height between presidents and their opponents.
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 standard error of the mean difference using the formula \( SE = \frac{s}{\sqrt{n}} \), where \( s \) is the standard deviation and \( n \) is the number of pairs.
5. Step 5: Construct the confidence interval for the mean difference using the formula \( \text{Confidence Interval} = \bar{x} \pm t \cdot SE \), where \( t \) is the critical value from the t-distribution for the desired confidence level (e.g., 95%) and degrees of freedom \( df = n-1 \). Analyze whether the interval includes 0 to determine the conclusion.

Key Concepts

Paired Sample Data

Paired sample data involves two related groups where each observation in one group is paired with a corresponding observation in the other group. This design is often used in studies to compare two conditions or treatments, allowing for a more accurate assessment of differences by controlling for variability between subjects.

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, the confidence interval provides insight into the significance of the results, as it indicates whether the null hypothesis can be rejected based on whether the interval includes the value of interest.

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 are often supported by confidence intervals, which can reinforce the conclusions drawn from the hypothesis test.