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).


a. Use the sample data with a 0.05 significance level to test the claim that for the population of heights of presidents and their main opponents, the differences have a mean greater than 0 cm.

Verified step by step guidance

1. Step 1: Calculate the differences between the heights of presidents and their main opponents for each pair. For example, for the first pair, the difference is 185 - 171 = 14 cm. Repeat this for all pairs in the dataset.
2. Step 2: Compute the mean of the differences obtained in Step 1. This involves summing all the differences and dividing by the number of pairs.
3. Step 3: Compute the standard deviation of the differences. Use the formula for standard deviation: \( \sigma = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} \), where \( x_i \) are the individual differences, \( \bar{x} \) is the mean of the differences, and \( n \) is the number of pairs.
4. Step 4: Perform a one-sample t-test to test the claim that the mean difference is greater than 0. The test statistic is calculated using \( t = \frac{\bar{x} - \mu}{s / \sqrt{n}} \), where \( \bar{x} \) is the sample mean, \( \mu \) is the hypothesized mean (0 in this case), \( s \) is the sample standard deviation, and \( n \) is the sample size.
5. Step 5: Compare the calculated t-value from Step 4 to the critical t-value at a significance level of 0.05 and degrees of freedom \( n-1 \). If the calculated t-value is greater than the critical t-value, reject the null hypothesis and conclude that the mean difference is greater than 0 cm.

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 heights of presidents and their main opponents are paired, allowing for a direct comparison of their heights. This method is useful for controlling variability and focusing on the differences between the two related samples.

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 the mean difference in heights is less than or equal to 0 cm, while the alternative hypothesis posits that the mean difference is greater than 0 cm. The significance level of 0.05 indicates the threshold for determining statistical significance.

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 heights are approximately normally distributed is crucial for applying certain statistical tests, such as the t-test, which relies on this property to make inferences about the population from the sample data.