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 for a difference in the two distributions when \(n=30\) and \(T^{+}=249\)

Short answer

Answer: We fail to reject the null hypothesis. There is not enough evidence to conclude that there is a difference between the two distributions at a significance level of 0.05.

Step-by-step solution

State the null and alternative hypotheses

The null hypothesis (\(H_0\)) is that there is no difference between the two distributions, while the alternative hypothesis (\(H_1\)) assumes that there is indeed a difference between them. So, the hypotheses can be stated as follows: $$ H_0 : D = 0 \\ H_1 : D \neq 0 $$ Where \(D\) represents the difference between the two distributions.

Calculate the test statistic

The test statistic for the Wilcoxon signed-rank test can be calculated by using the \(T^{+}\) (\(T\)-positive) value as given. However, we need to compute the \(T^{-}\) (\(T\)-negative) value as well: $$ T^{-} = n(n+1) / 2 - T^{+} $$ Given that \(n=30\) and \(T^{+} = 249\), we can plug these values into the formula: $$ T^{-} = 30(31) / 2 - 249 = 465 - 249 = 216 $$ Now, the test statistic (\(T\)) can be calculated as the smaller value of \(T^{+}\) and \(T^{-}\): $$ T = \min(T^{+}, T^{-}) = \min(249, 216) = 216 $$

Determine the rejection region and critical value

For a Wilcoxon signed-rank test, the rejection region for a given significance level \(\alpha\) can be found using the standard table or online calculator. In this case, we have \(\alpha=0.05\). Since we have the sample size \(n=30\) and \(\alpha=0.05\), we use the Wilcoxon signed-rank test table or online calculator to find the critical value for \(T\). Using the standard table or online calculator, we find the critical value for the two-tailed test to be \(T_{crit} = 131\), when \(n=30\) and \(\alpha= 0.05\).

State your conclusions

Now that we have the critical value \(T_{crit}=131\), we can compare it to our test statistic (\(T=216\)) to determine if we should reject or fail to reject the null hypothesis. Since \(T=216\) is greater than \(T_{crit}=131\), we fail to reject the null hypothesis. This means that we do not have enough evidence to conclude that there is a difference between the two distributions at a significance level of \(\alpha = 0.05\).

Key concepts

Null Hypothesis

The null hypothesis, denoted as \(H_0\), plays a pivotal role in statistical testing, including the Wilcoxon signed-rank test. It represents the default claim or assumption that there is no effect or no difference observed. In the context of a comparison between two related samples, the null hypothesis asserts that the median difference between the paired data is zero. Formally for the paired-difference experiment, \(H_0 : D = 0\) states that the two distributions being compared are the same. If the data do not provide sufficient evidence to challenge this claim, \(H_0\) remains accepted. Conversely, if the evidence suggests a significant difference, statisticians may reject \(H_0\) in favor of the alternative hypothesis.

Understanding \(H_0\) is crucial because it is the assumption under which the test statistic's distribution is evaluated. The decision to reject or not reject \(H_0\) hinges on whether the observed data is a rare event under this distribution, which translates to whether the observed effect is statistically significant or not.

Alternative Hypothesis

On the flip side of the null hypothesis, we have the alternative hypothesis, designated as \(H_1\) or \(H_a\). It directly contradicts \(H_0\) and represents the claim for which there is not yet evidence, but which is suspected or hoped to be true. In the case of the Wilcoxon signed-rank test, the alternative hypothesis might claim that there is a non-zero median difference between paired samples. It can take different forms depending on whether a one-tailed or two-tailed test is being performed. For the two-tailed test described in the exercise, the alternative hypothesis is \(H_1 : D eq 0\), suggesting that a significant difference does exist between the two distributions.

Identifying an appropriate \(H_1\) is important because it guides the direction and interpretation of the test. For example, if \(H_1\) suggested that the first distribution was greater than the second, a one-tailed test would be more appropriate. The type of alternative hypothesis affects not only the test conduct but also how we interpret the p-value or the test statistic's comparison to the rejection region.

Test Statistic

The test statistic is a single number that summarizes the data's evidence against the null hypothesis. It is specifically designed to quantify the strength of the evidence (or lack thereof) and is calculated from the sample data. In the Wilcoxon signed-rank test, the test statistic (\(T\)) is derived from the ranked differences between paired samples, taking into account their signs. Depending on the specific test variant, you might calculate \(T\) as the sum of ranks for the positive differences (\(T^+\)) or sum of ranks for the negative differences (\(T^-\)), or the lesser of the two sums, as in the provided exercise.

To compute the test statistic for our example, \(T^-\) was calculated and the minimum of \(T^+\) and \(T^-\) was taken as the test statistic. Through this process, \(T\) becomes the core value that we use to make inferences about our hypothesis. It is essentially the bridge between the sample data and the statistical decision: accept or reject \(H_0\).

Rejection Region

The rejection region is an integral part of hypothesis testing. It defines the range of values for which we would reject the null hypothesis \(H_0\) in favor of the alternative hypothesis \(H_1\). For any given significance level \(\alpha\), often 0.05, the rejection region corresponds to the tail(s) of the distribution of the test statistic under the null hypothesis that contains the most extreme \(\alpha\) fraction of the probability. In the Wilcoxon signed-rank test, we find the critical value \(T_{crit}\) that delineates the rejection region based on \(\alpha\) and the sample size (\(n\)).

If the computed test statistic falls within this rejection region, it indicates that such a result would be highly unlikely if the null hypothesis were true. Thus, seeing this result gives us reason to reject \(H_0\). In our case, even though we calculated a test statistic of 216, it was not within the rejection region determined by our critical value (131 for \(\alpha = 0.05\)) for \(n=30\). Therefore, we did not reject the null hypothesis. The concept of the rejection region makes the hypothesis test concrete, providing clear criteria for when to infer that results are statistically significant.