Question

In a study of hypnosis, breathing patterns were observed in an experimental group of subjects and in a control group. The measurements of total ventilation (liters of air per minute per square meter of body area) are shown. \({ }^{67}\) (These are the same data that were summarized in Exercise \(7.5 .6 .\) ) Use a Wilcoxon-Mann-Whitney test to compare the two groups at \(\alpha=0.10 .\) Use a nondirectional alternative. $$ \begin{array}{|cc|} \hline \text { Experimental } & \text { Control } \\ \hline 5.32 & 4.50 \\ 5.60 & 4.78 \\ 5.74 & 4.79 \\ 6.06 & 4.86 \\ 6.32 & 5.41 \\ 6.34 & 5.70 \\ 6.79 & 6.08 \\ 7.18 & 6.21 \\ \hline \end{array} $

Short answer

Calculate U for both groups, find the critical value of U from the table, and compare to determine if there is a statistically significant difference at the \(\alpha=0.10\) level.

Step-by-step solution

- Rank All Data

Combine both the experimental and control groups' data and rank them from smallest to largest, regardless of which group they belong to. Assign ranks to the measurements, with the smallest value assigned rank 1. In case of ties (identical values), assign the average rank that these values would have received if they were slightly different.

- Calculate Sum of Ranks for Each Group

Once all measurements have been ranked, add up the ranks for each group separately. Let the sum of the ranks for the experimental group be denoted as T_e and the sum for the control group as T_c.

- Use Test Statistic for Wilcoxon-Mann-Whitney Test

The test statistic for the Wilcoxon-Mann-Whitney test is usually denoted as U. Calculate U for each group using the following formulas: For the experimental group, U = T_e - (n_e * (n_e + 1))/2, where n_e is the number of observations in the experimental group. For the control group, U = T_c - (n_c * (n_c + 1))/2, where n_c is the number of observations in the control group. Choose the smaller of the two U values as the test statistic.

- Determine the Critical Value of U

Consult the Wilcoxon-Mann-Whitney critical values table for two samples of the given sizes at the desired alpha level. Compare your calculated test statistic to the critical value to determine whether the null hypothesis can be rejected.

- Make a Decision Based on the Critical Value

If the calculated U is less than or equal to the critical value from the table, reject the null hypothesis and conclude that there is a statistically significant difference between the two groups. If the calculated U is greater, do not reject the null hypothesis.

Key concepts

Nonparametric Statistics

Nonparametric statistics offer powerful tools for analyzing data without making assumptions about its distribution. Unlike parametric tests, which require the data to follow certain distributional criteria (e.g., normal distribution), nonparametric methods are distribution-free. These are particularly useful when working with small sample sizes, ordinal data, or when the presence of outliers would significantly skew the results of parametric tests.

Rank Sum Test

The rank sum test, also known as the Wilcoxon-Mann-Whitney test, is a nonparametric method used to determine whether there is a statistically significant difference between two independent groups. It is effective when comparing median values of two samples. To perform the test, all observations from both groups are ranked together. The ranks are then summed for each group separately, and the test statistic is calculated. This test does not rely on any assumptions about the populations from which the samples were drawn, making it a versatile tool in statistical analysis.

Hypothesis Testing

Hypothesis testing is a cornerstone of statistical analysis used to determine the evidence against a null hypothesis. The null hypothesis typically posits no effect or no difference, whereas the alternative hypothesis suggests there is an effect or a difference. In the context of the Wilcoxon-Mann-Whitney test, the null hypothesis would state that there is no difference in the median values between the two groups. By comparing a calculated test statistic to a critical value, researchers can decide whether to reject the null hypothesis or not, thus assessing if the observed data provide sufficient evidence to support the alternative hypothesis.

Statistical Significance

Statistical significance is a determination about the non-randomness of the results obtained from a statistical test. In simpler terms, it assesses whether the observed effect or difference is likely due to chance or if it reflects a true pattern in the data. A significance level, denoted as alpha (α), is predetermined by the researcher, often set at 0.05 or 0.10. If the probability of the observed test statistic is less than or equal to this alpha level, the result is considered statistically significant. This means that the null hypothesis is rejected in favor of the alternative hypothesis. It is essential to select an appropriate alpha level before conducting the test to avoid misuse of statistical analysis.