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. Mathematics Literacy Scores Through the Organization for Economic Cooperation and Development (OECD), 15-year-olds are tested in member countries in mathematics, reading, and science literacy. Listed are randomly selected total mathematics literacy scores (i.e. both genders) for selected countries in different parts of the world. Test, using the Kruskal-Wallis test, to see if there is a difference in means at \(\alpha=0.05\). $$ \begin{array}{ccc} \text { Western Hemisphere } & \text { Europe } & \text { Eastern Asia } \\ \hline 527 & 520 & 523 \\ 406 & 510 & 547 \\ 474 & 513 & 547 \\ 381 & 548 & 391 \\ 411 & 496 & 549 \end{array} $$

Short answer

Test the differences in medians across groups using the Kruskal-Wallis test steps.

Step-by-step solution

State the Hypotheses and Identify the Claim

The null hypothesis (\(H_0\)) states that there is no difference in the mathematics literacy score medians among the groups (Western Hemisphere, Europe, Eastern Asia). Thus, \(H_0: M_1 = M_2 = M_3\) where \(M_i\) represents the median scores for each group. The alternative hypothesis (\(H_a\)) suggests that at least one group's median score is different. We are testing whether any differences among medians exist, making this the claim: there is a difference in the mathematics literacy score medians.

Arrange Data and Rank

Combine all the data into a single list to rank the scores regardless of group. The scores are 527, 406, 474, 381, 411, 520, 510, 513, 548, 496, 523, 547, 547, 391, 549. Rank them from the lowest to the highest, assigning an average rank for ties (e.g., ranks for 547 appear twice).

Calculate Group Rank Sums

Sum the ranks for each group based on their original data points. For Western Hemisphere, sum the ranks of its scores: 527, 406, 474, 381, 411. Repeat for Europe: 520, 510, 513, 548, 496. Finally, repeat for Eastern Asia: 523, 547, 547, 391, 549.

Calculate the Kruskal-Wallis Test Statistic

Use the formula: \[ H = \frac{12}{N(N+1)} \sum \frac{R_i^2}{n_i} - 3(N+1) \]where \(N\) is the total number of observations across all groups, \(R_i\) is the rank sum for group \(i\), and \(n_i\) is the number of observations in group \(i\). Substitute the calculated rank sums and group sizes into this formula to compute \(H\).

Determine the Critical Value

For the Kruskal-Wallis test with 3 groups, the degrees of freedom \(df = k - 1\), where \(k\) is the number of groups. Here, \(df = 3 - 1 = 2\). Use a chi-square distribution table to find the critical value for \(df = 2\) and \(\alpha = 0.05\), which is approximately 5.991.

Compare Test Statistic and Make Decision

Compare the calculated test statistic \(H\) with the critical value. If \(H\) is greater than 5.991, reject the null hypothesis \(H_0\). Otherwise, fail to reject \(H_0\).

Summarize the Results

If the null hypothesis is rejected, conclude that there is enough evidence to suggest a difference in the median mathematics literacy scores among at least one of the groups. If it is not rejected, conclude that there is not enough evidence to suggest a difference appears.

Key concepts

Hypothesis Testing

Hypothesis testing is a structured procedure used to make statistical decisions based on experimental data. The goal is to determine whether there is enough evidence to support a specific hypothesis about a population parameter.

In the context of the Kruskal-Wallis test, hypothesis testing begins with formulating the null and alternative hypotheses:

• Null Hypothesis (\(H_0\)): Assumes that there is no difference in the median scores across groups. Essentially, \(M_1 = M_2 = M_3\).
• Alternative Hypothesis (\(H_a\)): Suggests that at least one group's median score is different, indicating a variation among groups.

The null hypothesis is subjected to a test to statistically infer whether it holds true or should be rejected in favor of the alternative hypothesis. This process involves evaluating the test statistic in comparison to a critical value, which helps in making an objective decision.

Hypothesis testing follows a standard methodology:

• Define null and alternative hypotheses.
• Determine the critical value based on a chosen significance level \(\alpha\).
• Calculate the test statistic from the data.
• Make a decision to accept or reject the null hypothesis.
• Summarize findings to conclude whether the hypothesis is supported or not.

Critical Value

The critical value is a threshold that helps determine the fate of the null hypothesis during hypothesis testing. For the Kruskal-Wallis test, the critical value depends on the chi-square distribution.

To find the critical value, consider:

• Degrees of Freedom (\(df\)): The number of groups minus one. In this example, with three groups (Western Hemisphere, Europe, Eastern Asia), the degrees of freedom is calculated as \(df = 3 - 1 = 2\).
• Significance Level (\(\alpha\)): Typically set at 0.05 for a 5% risk of rejecting the true null hypothesis, known as a Type I error.

With the degrees of freedom and significance level in hand, use a chi-square distribution table to find the critical value. For \(df = 2\) and \(\alpha = 0.05\), the critical value is approximately 5.991.

The decision rule states:

• If the test statistic \(H\) exceeds the critical value, reject the null hypothesis \(H_0\).
• If \(H\) is less, fail to reject \(H_0\).

This comparison provides the basis for deciding whether observed differences among groups are statistically significant.

Rank Sums

Rank sums are critical in non-parametric tests like the Kruskal-Wallis test, where actual data values are replaced by their ranks. This process helps in dealing with non-normal data distributions.

To calculate rank sums:

• Combine all group data into a single dataset.
• Assign ranks to these combined data points, from lowest to highest. Ties are given average ranks.
• Sum these ranks for each separate group to determine the rank sums \(R_i\).

Consider the provided example:
- Western Hemisphere scores are ranked, and their ranks are summed.- Similarly, calculate for Europe and Eastern Asia.

Rank sums result in a summary measure representing each group's data, useful for computing the Kruskal-Wallis test statistic \(H\). This method ensures that the central tendencies are unbiased and resistant to outliers.
The rank sums help assess whether the differences among group medians are statistically notable.

Chi-Square Distribution

The chi-square distribution is an essential component of hypothesis testing, particularly for non-parametric tests like Kruskal-Wallis. It's used for assessing how well experimental data fit expected distributions.

Key aspects of chi-square distribution include:

• Nature: Skewed to the right, but becomes more symmetric with more degrees of freedom.
• Applications: Used in significance testing, such as comparing observed and expected frequencies in categorical data.
• Degrees of Freedom (\(df\)): Increases with the number of categories or groups minus one. More \(df\) results in a shift towards normality.

In the Kruskal-Wallis test, the test statistic \(H\) calculated from rank sums is evaluated against the chi-square distribution. This is because under the null hypothesis, \(H\) follows approximately a chi-square distribution.

Using the chi-square value obtained from tables, one compares it with the calculated \(H\):

• If \(H\) is greater than the chi-square critical value, the findings support that differences among groups may not be due to chance.
• If not, there's inadequate evidence to claim significant differences.

This statistical tool is invaluable for deriving conclusions from group comparisons based on ranks.