Question

The information in Exercises 5-6 refers to a paired-difference experiment. Analyze the data using the Wilcoxon signed-rank test. State the null and alternative hypotheses to be tested and calculate the test statistic. Find the rejection region for \(\alpha=.05\) and state your conclusions. \(\left[\right.\) NOTE \(\left.: T^{+}+T^{-}=n(n+1) / 2 .\right]\) Test whether distribution 1 lies to the right of distribution 2 when \(n=30\) and \(T^{+}=249\).

Short answer

Provide a brief explanation.

Step-by-step solution

State the null and alternative hypotheses

The null hypothesis (\(H_0\)) and alternative hypothesis (\(H_a\)) are as follows: \(H_0\): Distribution 1 does not lie to the right of distribution 2. \(H_a\): Distribution 1 lies to the right of distribution 2.

Calculate the test statistic \(T^{-}\)

We are given \(T^{+} = 249\) and the formula \(T^{+} + T^{-} = n(n+1)/2\). We can use this formula to calculate the value of \(T^{-}\). Given \(n = 30\), we can calculate \(n(n+1)/2\): \(30(30 + 1)/2 = 30(31)/2 = 465\) Now we can calculate \(T^{-}\): \(T^{-} = 465 - 249 = 216\)

Find the critical value for \(\alpha = 0.05\)

For a sample size of \(n = 30\) and a significance level of \(\alpha = 0.05\), we can look up the critical value \(T_c\) in a Wilcoxon signed-rank test table. The critical value found in the table is \(T_c = 144\).

Determine the rejection region and state the conclusion

The rejection region is determined by comparing the calculated test statistic \(T^{-}\) to the critical value \(T_c\). If \(T^{-} \leq T_c\), we reject the null hypothesis in favor of the alternative hypothesis. Otherwise, we fail to reject the null hypothesis. In this case, \(T^{-} = 216 > T_c = 144\). So, we fail to reject the null hypothesis, and we cannot conclude that distribution 1 lies to the right of distribution 2.

Key concepts

Null and Alternative Hypotheses

Understanding the null and alternative hypotheses is crucial in any statistical test, including the Wilcoxon signed-rank test. In hypothesis testing, the null hypothesis (\(H_0\)) represents a default position that there is no effect or no difference. It’s what we assume to be true before collecting any data. For the example given, the null hypothesis states that Distribution 1 does not lie to the right of Distribution 2, implying no shift or tendency of one distribution to have higher values than the other.

The alternative hypothesis (\(H_a\text{ or }H_1\text{ depending on notation preference}\text{) contradicts the null hypothesis and is what we hope to support with the data}. In the case of the Wilcoxon signed-rank test, the alternative hypothesis claims that Distribution 1 lies to the right of Distribution 2, suggesting a tendency for values in Distribution 1 to be greater than those in Distribution 2. Determining these hypotheses is the first step in the process of conducting a statistical test.

Test Statistic Calculation

The test statistic is a crucial component that helps us decide whether to reject the null hypothesis. In the Wilcoxon signed-rank test, it involves calculating the differences between paired observations, ranking these differences in order of their absolute values, and then summing the ranks corresponding to the positive and negative differences.

In our example, we were provided with the sum of the ranks for the positive differences, denoted as \(T^{+}\), and given the value of 249. To find the sum of the ranks of the negative differences, \(T^{-}\), we use the provided relation \(T^{+} + T^{-} = n(n + 1) / 2\). Here, \(n\) is the sample size, which is 30. After performing the calculation, we obtained \(T^{-} = 216\), which is our test statistic and the key figure in determining the outcome of our hypothesis test.

Rejection Region

The rejection region is the range of values for which the null hypothesis is not considered plausible and is therefore 'rejected'. It's defined in terms of a significance level, denoted by \(\alpha\), which is the probability of making the mistake of rejecting the null hypothesis when, in fact, it's true (also known as a Type I error).

For our example with an \(\alpha = 0.05\) and sample size of \(n = 30\), we find a critical value \(T_c = 144\) from the Wilcoxon signed-rank table; this separates the rejection region from the non-rejection region. In simpler terms, if our test statistic \(T^{-}\) is less than or equal to 144, we would reject the null hypothesis. As our \(T^{-}\) is 216, it does not fall into the rejection region, so we do not reject the null hypothesis.

Non-parametric Statistical Test

The Wilcoxon signed-rank test is a non-parametric statistical test, which means it doesn't assume the data come from a particular distribution, like the normal distribution, which is a common assumption in many other statistical tests. These types of tests are especially valuable when handling small sample sizes, skewed data, or data with outliers.

Non-parametric tests use the median as a measure of central tendency rather than the mean, and they assess the ordinal or rank information in the data. This makes them less sensitive to extreme values or non-normality. The Wilcoxon signed-rank test is appropriate when dealing with paired samples or matched sets and is a robust alternative to the paired sample t-test, which requires normally distributed differences. In applying this test to our problem, we avoid the potential pitfalls of parametric assumptions and use a method that aligns better with the nature of our data.