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.


c. What do you conclude about the Freshman 15 belief?

Verified step by step guidance

1. Step 1: Calculate the differences between the paired weights for September and April. For each individual, subtract the September weight from the April weight to find the change in weight.
2. Step 2: Compute the mean of the differences. Add all the differences together and divide by the number of paired samples to find the average change in weight.
3. Step 3: Calculate 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 differences, \( \bar{x} \) is the mean of the differences, and \( n \) is the number of paired samples.
4. Step 4: Perform a hypothesis test to determine if the mean difference is significantly different from 15 lb (or 6.8 kg). Use a t-test for paired samples, with the null hypothesis \( H_0: \mu = 6.8 \) and the alternative hypothesis \( H_a: \mu \neq 6.8 \). Calculate the t-statistic using \( t = \frac{\bar{x} - \mu}{s / \sqrt{n}} \), where \( \bar{x} \) is the mean difference, \( \mu \) is the hypothesized mean, \( s \) is the standard deviation, and \( n \) is the sample size.
5. Step 5: Compare the calculated t-statistic to the critical t-value from the t-distribution table at the chosen significance level (e.g., \( \alpha = 0.05 \)). If the t-statistic falls outside the critical range, reject the null hypothesis. Based on the results, draw a conclusion about the validity of the Freshman 15 belief.

Key Concepts

Paired Sample Data

Paired sample data involves two related groups where measurements are taken from the same subjects at two different times or under two 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.

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 weights have an approximately normal distribution is crucial for applying statistical tests that rely on this property, such as the paired t-test.

Statistical Inference

Statistical inference involves drawing conclusions about a population based on sample data. In this case, the analysis of the weight differences will help determine whether the belief in the 'Freshman 15'—that students gain 15 pounds during their freshman year—is supported by the data collected from the sample of students.