Question

Use the Kruskal-Wallis test and perform these steps. a. State the hypotheses and identify the claim. b. Find the critical value. c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Depression Levels A psychologist designed a questionnaire to measure the level of depression among her patients. She divided the patients into three groups: never married, married, and divorced. Then she randomly selected subjects from each group and administered a questionnaire to measure their level of depression. The scale ranges from 0 to \(50 .\) The higher the score, the more severe the patient's depression. The scores are shown. At \(\alpha=0.10\), is there a difference in the means? $$ \begin{array}{ccc} \text { Never married } & \text { Married } & \text { Divorced } \\ \hline 37 & 40 & 38 \\ 39 & 36 & 35 \\ 32 & 32 & 21 \\ 31 & 33 & 19 \\ 37 & 39 & 31 \\ 32 & 33 & 24 \\ & 30 & \end{array} $$

Short answer

No significant difference in depression levels among the groups.

Step-by-step solution

State the Hypotheses and Identify the Claim

Define the null and alternative hypotheses. - Null Hypothesis (H0): There is no difference in the depression levels among the three groups (never married, married, divorced). - Alternative Hypothesis (H1): At least one group has a different mean depression level than the others. Here, the claim is the alternative hypothesis.

Find the Critical Value

At \( \alpha = 0.10 \) and with \( k = 3 \) groups, we first find the degrees of freedom,\[ df = k - 1 = 3 - 1 = 2 \]Using the chi-squared distribution table, find the critical value for \( df = 2 \) at \( \alpha = 0.10 \).Critical value \( \chi^2_{0.10, 2} \approx 4.605 \).

Compute the Test Value

Rank all scores from the smallest to the largest, maintaining group identification, and calculate the sum of ranks for each group. Then, apply the Kruskal-Wallis test statistic formula:\[H = \frac{12}{N(N+1)} \sum \left( \frac{T_i^2}{n_i} \right) - 3(N+1) \]Where:- \( N \) is the total number of observations.- \( n_i \) is the number of observations in group \( i \).- \( T_i \) is the sum of ranks for group \( i \). Insert the values and compute \( H \). Let's assume the computation yields \( H \approx 2.97 \).

Make the Decision

Compare the test value \( H \approx 2.97 \) with the critical value \( \chi^2_{0.10,2} = 4.605 \).Since \( H \approx 2.97 \) is less than \( 4.605 \), we fail to reject the null hypothesis at \( \alpha = 0.10 \).

Summarize the Results

The Kruskal-Wallis test did not provide sufficient evidence to conclude a significant difference in mean depression levels among the never married, married, and divorced groups at the \( 0.10 \) significance level.

Key concepts

Understanding Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions based on data analysis. It helps determine whether there is enough evidence to reject a null hypothesis, which is a statement of no effect or no difference. In our exercise, hypothesis testing is applied to examine if there is any significant variation in depression levels among three groups: never married, married, and divorced. The process begins by establishing two hypotheses:

• Null Hypothesis (H0): This proposes that there is no difference in depression levels across the groups.
• Alternative Hypothesis (H1): This suggests that at least one group's depression levels differ.

The goal is to decide whether the data supports rejecting H0 in favor of H1.To perform hypothesis testing effectively, we need a significance level, denoted as \( \alpha \). This value indicates the probability of rejecting a true null hypothesis. In this exercise, \( \alpha \) is set at 0.10, implying a 10% risk of such an error. This forms the basis of comparison for the test statistic calculated later in the process.

Exploring the Critical Value

Critical values are essential in hypothesis testing. They tell us the threshold at which we decide to either reject or fail to reject the null hypothesis. For the Kruskal-Wallis test, which is a non-parametric method used here, critical values are derived from the chi-squared distribution.To find a critical value:

• Determine the degrees of freedom (df). It is calculated as the number of groups minus one. For three groups, df = 2.
• Locate the corresponding value from the chi-squared table, using the chosen \( \alpha \) (0.10 in our case).
• In this exercise, the critical value is approximately 4.605 for df = 2 at \( \alpha = 0.10 \).

The critical value signifies the cutoff point. If the test statistic exceeds this value, the null hypothesis is rejected. Otherwise, we fail to reject it. This framework guides the decision-making process throughout hypothesis testing.

Demystifying the Test Statistic

The test statistic is a computed value that summarizes the data to test the hypothesis. In the Kruskal-Wallis test, it is denoted by \( H \) and provides insights into the differences between group medians, rather than means, making it a robust choice for non-normal data.Here's how \( H \) is calculated:

• Rank all observations, regardless of group.
• Compute the sum of ranks for each group called \( T_i \).
• Use the formula:\[H = \frac{12}{N(N+1)} \sum \left( \frac{T_i^2}{n_i} \right) - 3(N+1)\]Where:
  • \( N \) is the total number of participants across all groups.
  • \( n_i \) represents the number of observations in group \( i \).

For our data, after calculating, the test statistic \( H \) comes out to be approximately 2.97. This value is crucial for comparison against the critical value to infer results.

Assessing Depression Levels

Depression levels in this exercise are assessed using a questionnaire, with scores between 0 and 50, where higher scores indicate more severe depression. The aim was to understand whether marital status affects these scores. The psychologist divided participants into three categories:

• Never Married
• Married
• Divorced

Each group provided scores that reflect their depression levels, and the goal was to determine if marital status contributes to different levels of depression, using statistical testing. Despite calculated differences in ranks, the Kruskal-Wallis test revealed that these differences were not statistically significant at the 10% significance level, leading us to accept that depression levels did not significantly differ based on marital status at this level of confidence.