In Exercises 1–10, use a 0.05 significance level with the indicated test. If no particular test is specified, use the appropriate nonparametric test from this chapter.




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. Use the sign test to test the claim that for the population of freshman male college students there is not a significant difference between the weights in September and the weights in the following April. What do you conclude about the Freshman 15 belief?

Verified step by step guidance

1. Step 1: Understand the problem and identify the test. The problem requires using the sign test to determine if there is a significant difference between the weights of male college freshmen in September and April. The null hypothesis (H₀) states that there is no significant difference in weights, while the alternative hypothesis (H₁) states that there is a significant difference.
2. Step 2: Calculate the differences between the paired weights for September and April. For each pair, subtract the September weight from the April weight. Record whether the difference is positive, negative, or zero.
3. Step 3: Count the number of positive differences, negative differences, and zeros. Zeros are excluded from the analysis as they do not contribute to the sign test.
4. Step 4: Use the sign test formula to calculate the test statistic. The sign test is based on the binomial distribution, where the number of positive differences is compared to the expected number under the null hypothesis. The significance level is 0.05.
5. Step 5: Compare the test statistic to the critical value or use the p-value approach. If the test statistic exceeds the critical value or the p-value is less than 0.05, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Interpret the results in the context of the Freshman 15 belief.

Key Concepts

Sign Test

The sign test is a nonparametric statistical method used to evaluate the median of a single sample or the differences between paired samples. It focuses on the direction of changes (positive or negative) rather than their magnitude, making it suitable for ordinal data or when the assumptions of parametric tests are not met. In this context, it will help determine if there is a significant difference in weights of male freshmen from September to April.

Significance Level

The significance level, often denoted as alpha (α), is the threshold for determining whether a statistical result is significant. In this case, a significance level of 0.05 indicates that there is a 5% risk of concluding that a difference exists when there is none (Type I error). This level is commonly used in hypothesis testing to assess the strength of evidence against the null hypothesis.

Null Hypothesis

The null hypothesis (H0) is a statement that there is no effect or no difference, serving as a starting point for statistical testing. In this scenario, the null hypothesis posits that there is no significant difference in the weights of male freshmen between September and April. Testing this hypothesis allows researchers to determine if the observed data provides enough evidence to reject it in favor of an alternative hypothesis.