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.


Wilcoxon Signed-Ranks Test Repeat the preceding exercise using the Wilcoxon signed-ranks test.

Verified step by step guidance

1. Step 1: Understand the Wilcoxon Signed-Ranks Test. This is a nonparametric test used to compare paired data. It evaluates whether the median difference between pairs of observations is zero, making it an alternative to the paired t-test when the data does not meet the normality assumption.
2. Step 2: Calculate the differences between the paired observations. For each pair, subtract one value from the other (e.g., difference = observation1 - observation2). Ignore the sign for now, as we will rank the absolute differences.
3. Step 3: Rank the absolute differences. Assign ranks to the absolute values of the differences, starting with the smallest difference as rank 1. If there are tied differences, assign the average rank to each tied value.
4. Step 4: Reapply the signs to the ranks. For each difference, assign the corresponding rank a positive or negative sign based on whether the original difference was positive or negative.
5. Step 5: Compute the test statistic. Sum the positive ranks (T+) and the negative ranks (T-). The test statistic is the smaller of these two sums. Compare this test statistic to the critical value from the Wilcoxon Signed-Ranks Test table for the given sample size and significance level (α = 0.05). If the test statistic is less than or equal to the critical value, reject the null hypothesis; otherwise, fail to reject it.

Key Concepts

Wilcoxon Signed-Ranks Test

The Wilcoxon Signed-Ranks Test is a nonparametric statistical test used to compare two related samples or matched observations. It assesses whether their population mean ranks differ, making it suitable for situations where the data does not meet the assumptions of normality required for parametric tests. This test is particularly useful for small sample sizes or ordinal data.

Significance Level

The significance level, often denoted as alpha (α), is the threshold for determining whether a statistical result is 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 tests that do not assume a specific distribution for the data, making them ideal for analyzing ordinal data or data that violate the assumptions of parametric tests. These tests, like the Wilcoxon Signed-Ranks Test, are robust to outliers and can be used when sample sizes are small or when the data is not normally distributed, providing flexibility in statistical analysis.