Question

Use the Wilcoxon rank sum test to determine whether population 1 lies to the left of population 2 by (1) stating the null and alternative hypotheses to be tested, (2) calculating the values of \(T_{1}\) and \(T_{l}^{*},(3)\) finding the rejection region for \(\alpha=.05,\) and (4) stating your conclusions. $$ \begin{array}{l|lllll} \text { Sample } 1 & 1 & 3 & 2 & 3 & 5 \\ \hline \text { Sample } 2 & 4 & 7 & 6 & 8 & 6 \end{array} $$

Short answer

A) There is insufficient evidence to determine the relationship between the two populations. B) Population 1 lies to the right of population 2. C) Population 1 lies to the left of population 2. D) Both populations are identical.

Step-by-step solution

State the null and alternative hypotheses

The null hypothesis (H0) states that there is no difference between the two populations; that is, they have the same probability distribution. $$ H_{0}: P(X_{1} < X_{2}) = P(X_{1} > X_{2}) $$ The alternative hypothesis (H1) states that population 1 lies to the left of population 2 (i.e., population 1 has a lower probability distribution than population 2). $$ H_{1}: P(X_{1} < X_{2}) > P(X_{1} > X_{2}) $$

Calculate the values of \(T_{1}\) and \(T_{l}^{*}\)

First, combine both samples and rank them in ascending order: $$ \begin{array}{c|c|c} \text { Rank } & \text { Sample } 1 & \text { Sample } 2 \\\ \hline 1 & 1 & \\\ 2 & 2 & \\\ 3 & \ & \\\ 4 & \ & 4 \\\ 5 & 3 & \\\ 6 & \ & 6 \\\ 7 & 5 & 7 \\\ 8 & \ & 8 \\\ 9 & \ & 6 \end{array} $$ Now, calculate the sum of the ranks for Sample 1: $$T_{1}=1+2+5=8$$ Next, find the expected value of \(T_{1}^{*}\) and variance: $$E(T_{1}^{*})=\frac{n_{1}(n_{1}+n_{2}+1)}{2}=\frac{5(5+5+1)}{2}=\frac{5(11)}{2}=27.5$$ $$Var(T_{1}^{*})=\frac{n_{1}n_{2}(n_{1}+n_{2}+1)}{12}=\frac{(5)(5)(11)}{12}=\frac{275}{12}$$ Now, calculate the lower limit of the standardized test statistic: $$T_{l}^{*}=\frac{T_{1}-E(T_{1}^{*})}{\sqrt{Var(T_{1}^{*})}}=\frac{8-27.5}{\sqrt{\frac{275}{12}}}≈-3.36$$

Find the rejection region for \(\alpha=.05\)

The rejection region is defined as the set of all values of the test statistic, \(T_{l}^{*}\), for which we reject the null hypothesis. Since we are performing a one-tailed test and \(\alpha = 0.05\), we will compare the calculated \(T_{l}^{*}\) to the critical value \(z_{\alpha} = -1.645\) (from the standard normal distribution table). If \(T_{l}^{*} < z_{\alpha}\), we reject the null hypothesis.

State the conclusions

Our calculated value of the test statistic, \(T_{l}^{*} ≈ -3.36\), is less than the critical value \(z_{\alpha} = -1.645\). Therefore, we reject the null hypothesis and conclude that there is sufficient evidence to support the alternative hypothesis that population 1 lies to the left of population 2.

Key concepts

Null and Alternative Hypotheses

Understanding the null and alternative hypotheses is crucial in any statistical test, including the Wilcoxon rank sum test. The null hypothesis (\textbf{H0}) serves as a default statement that indicates no effect or no difference. It's the claim we aim to challenge or verify. For the Wilcoxon test, \textbf{H0} asserts that two independent samples come from identical populations with the same probability distribution.

In contrast, the alternative hypothesis (\textbf{H1}) is what researchers suspect to be true. It represents a new claim about the populations, such as one distribution being shifted to the left or to the right of the other. In the context of our exercise, \textbf{H1} suggests that population 1 has a tendency to generate lower values than population 2. This is expressed mathematically as \textbf{H1}: \( P(X_{1} < X_{2}) > P(X_{1} > X_{2}) \).

If the test results lead to rejection of \textbf{H0}, it supports the alternative hypothesis, suggesting the existence of a significant difference between the populations. However, failing to reject the null hypothesis does not confirm its truth; it merely indicates insufficient evidence to favor \textbf{H1}. The way we shape these hypotheses has a profound impact on the analysis and interpretation of data.

Nonparametric Statistics

Nonparametric statistics come into play when the assumptions necessary for parametric tests are not satisfied. These could include situations where the sample sizes are small, the data are not normally distributed, or the type of data doesn't allow for parametric analysis, such as ordinal data or ranks.

The Wilcoxon rank sum test, a key player in nonparametric statistical methods, does not assume a specific probability distribution for the data. It is employed when we cannot confirm that the data follow a normal distribution or when measuring a median is more appropriate than a mean. In the given exercise, the Wilcoxon rank sum test essentially compares the medians of two independent samples to test the hypothesis of interest.

Because nonparametric tests like the Wilcoxon are based on ranks rather than data values, they are less sensitive to outliers and can offer a robust alternative to parametric tests, which might be inapplicable or misleading when their strict assumptions are not met. The robustness and flexibility of nonparametric methods make them indispensable tools in statistical analysis.

Probability Distributions

Probability distributions are fundamental to understanding statistical tests. A probability distribution describes how the values of a random variable are distributed. There are many types of probability distributions, each with their own properties, which describe different kinds of data and situations. For parametric tests, certain probability distributions such as the normal distribution are assumed.

In nonparametric tests like the Wilcoxon rank sum test, however, we do not assume any specific probability distribution. The central focus is on the rank ordering of the data rather than the actual data values. This approach mitigates the impact of the shape of the probability distribution on the analysis. Despite not knowing the underlying distribution, we can still obtain useful information about the population by comparing the rank sums. Through this lens, we can analyze a variety of data types and draw conclusions without the need for distributional assumptions.

As statistics often deal with uncertainty and variability, mastering the concept of probability distributions is essential. It aids in choosing the correct statistical test and interpreting the results accurately, whether the distribution is known and parametric methods are used, or it remains unspecified as is necessary for nonparametric techniques.