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.


The Freshman 15 The “Freshman 15” refers to the belief that college students gain 15 lb (or 6.8 kg) during their freshman year. Listed below are weights (kg) of randomly selected male college freshmen (from Data Set 13 “Freshman 15” in Appendix B). The weights were measured in September and later in April.


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 weights for September and April. For each pair, subtract the September weight from the April weight to find the difference (April - September).
2. Step 2: Compute the mean of the differences. Add all the differences together 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 standard error of the mean difference using the formula \( SE = \frac{s}{\sqrt{n}} \), where \( s \) is the standard deviation of the differences and \( n \) is the number of pairs.
5. Step 5: Construct the confidence interval using the formula \( \bar{x} \pm t \cdot SE \), where \( \bar{x} \) is the mean difference, \( t \) is the critical t-value from the t-distribution table for the desired confidence level, and \( SE \) is the standard error. Interpret the confidence interval to determine whether it includes 0, which would lead to the same conclusion as the hypothesis test in part (a).

Key Concepts

Paired Sample Data

Paired sample data involves two related groups where measurements are taken from the same subjects at different times or under different conditions. In this context, the weights of male college freshmen are measured in September and again in April, allowing for a direct comparison of weight changes over time.

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 this case, constructing a confidence interval for the weight differences will help assess whether the average weight gain aligns with the hypothesis of the 'Freshman 15.'

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about population parameters based on sample data. In this scenario, the null hypothesis might state that there is no significant weight gain among freshmen, while the alternative hypothesis suggests that there is a significant gain. The results from the confidence interval can support or refute this hypothesis.