The Newman-Keuls test is a statistical procedure used to determine significant differences between means in a one-way or two-way analysis of variance (ANOVA). It is a post-hoc test, meaning it is used to make comparisons after the initial ANOVA has been conducted.

One example of the Newman-Keuls test might be to compare the average scores of three different groups on a test. Group A consists of students who received extra tutoring, group B consists of students who received no extra help, and group C consists of students who received a different type of assistance. After administering the test, the mean scores for each group are calculated. The ANOVA is conducted to determine if there are significant differences between the means of the three groups. If the ANOVA shows that there are significant differences, the Newman-Keuls test can be used to determine which groups differ significantly from one another.

In this example, the Newman-Keuls test would compare the mean scores of group A to group B, group A to group C, and group B to group C. If the mean score for group A is significantly higher than the mean score for group B, it can be concluded that the extra tutoring provided to group A had a positive impact on their test scores. If the mean score for group C is significantly higher than the mean score for group B, it can be concluded that the type of assistance provided to group C had a positive impact on their test scores.

Another example of the Newman-Keuls test might be to compare the effectiveness of two different types of therapy in reducing anxiety in a group of individuals. Group A receives therapy type A, while group B receives therapy type B. After the therapy sessions are completed, the levels of anxiety are measured and the mean scores for each group are calculated. The ANOVA is conducted to determine if there are significant differences between the means of the two groups. If the ANOVA shows that there are significant differences, the Newman-Keuls test can be used to determine which group had a significantly lower level of anxiety.

In this example, the Newman-Keuls test would compare the mean anxiety scores of group A to group B. If the mean anxiety score for group A is significantly lower than the mean anxiety score for group B, it can be concluded that therapy type A was more effective in reducing anxiety levels.

The Newman-Keuls test is considered a conservative test, meaning that it has a lower risk of committing a Type I error (falsely rejecting the null hypothesis). However, it also has a higher risk of committing a Type II error (failing to reject the null hypothesis when it should be rejected). As such, it may be less powerful in detecting significant differences between means compared to other post-hoc tests such as the Tukey or Bonferroni tests.

In conclusion, the Newman-Keuls test is a statistical procedure used to determine significant differences between means in a one-way or two-way ANOVA. It is used to make comparisons between groups after the initial ANOVA has been conducted and can be applied to a variety of situations, such as comparing test scores or the effectiveness of different therapies. It is considered a conservative test with a lower risk of committing a Type I error, but a higher risk of committing a Type II error.