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.


Rank Correlation Use the paired sample data from Exercise 1 and analyze the data using the rank correlation coefficient. What does the result tell us about the belief that college students gain 15 lb (or 6.8 kg) during their freshman year?

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to analyze the paired sample data using the rank correlation coefficient (Spearman's rank correlation coefficient). This test is used to measure the strength and direction of the association between two ranked variables. The significance level is 0.05.
2. Step 2: Rank the data. For each variable in the paired data, assign ranks to the values. If there are tied values, assign the average of the ranks that would have been assigned to those values.
3. Step 3: Calculate the difference in ranks (d) for each pair of observations. For each pair, subtract the rank of one variable from the rank of the other variable.
4. Step 4: Compute the squared differences (d²) for each pair. Then, sum up all the squared differences to get Σd².
5. Step 5: Use the formula for Spearman's rank correlation coefficient: $r\_s\; =\; 1\; -\; (6\Sigma d²)\; /\; (n(n²\; -\; 1))$, where n is the number of pairs. Compare the calculated r_s value to the critical value for Spearman's rank correlation at the 0.05 significance level to determine if there is a significant correlation. Interpret the result in the context of the belief about weight gain during the freshman year.

Key Concepts

Rank Correlation Coefficient

The rank correlation coefficient, often represented by Spearman's rho, measures the strength and direction of association between two ranked variables. It assesses how well the relationship between the variables can be described using a monotonic function. This is particularly useful when the data does not meet the assumptions of parametric tests, making it a robust choice for nonparametric analysis.

Significance Level

The significance level, commonly denoted as alpha (α), is the threshold for determining whether a result is statistically significant. In this context, 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). It helps researchers decide whether to reject the null hypothesis based on the p-value obtained from the test.

Nonparametric Tests

Nonparametric tests are statistical methods that do not assume a specific distribution for the data, making them suitable for ordinal data or when sample sizes are small. These tests are particularly useful when the assumptions of parametric tests, such as normality, are violated. Examples include the Wilcoxon signed-rank test and the Kruskal-Wallis test, which are often used in place of t-tests or ANOVA when data does not meet required conditions.