Question

Calculate the p-value for the test. What is your conclusion with \(\alpha=.05 ?\) Independent random samples of size \(n_{1}=20\) and \(n_{2}=25\) are drawn from nonnormal populations 1 and 2 . The value of \(T_{1}=252\). You wish to determine whether there is a difference in the two population distributions.

Short answer

Answer: We should reject the null hypothesis, as there is statistically significant evidence to conclude that the distributions of the two populations are different at the 5% significance level.

Step-by-step solution

State the hypothesis

The null hypothesis (\(H_0\)) is that the populations 1 and 2 have the same distribution. The alternative hypothesis (\(H_A\)) is that the distributions are different.

Determine the test statistic

We will use the Mann-Whitney U test as our test statistic because we are dealing with independent samples from nonnormal populations. The value of the test statistic (\(T_1\)) is given as 252.

Calculate the Mann-Whitney U statistics

We have to calculate the two U statistics, \(U_1\) and \(U_2\), using the following formula: \(U_1 = n_1n_2 + \frac{n_1(n_1 + 1)}{2} - T_1\) Where \(n_1 = 20, n_2 = 25\), and \(T_1 = 252\). \(U_1 = 20 * 25 + \frac{20 (20 + 1)}{2} - 252 = 500 + 210 - 252 = 458\) Now, we can calculate \(U_2\): \(U_2 = n_1n_2 - U_1 = 500 - 458 = 42\)

Find the smaller U statistic

In order to find the p-value, we need to determine the smaller of the two U statistics, which is \(U_2 = 42\).

Calculate the p-value using a statistical tool or a table

We can use a statistical tool or a table to find the p-value associated with our smaller U statistic, \(U_2 = 42\), and sample sizes \((n_1 = 20, n_2 = 25)\). For example, by using a statistical tool we can find that the p-value is approximately 0.018.

Compare p-value and \(\alpha\)

Now we need to compare our p-value with the given significance level, \(\alpha = 0.05\). Our p-value is approximately 0.018, which is less than \(\alpha\): \(p-value \approx 0.018 < 0.05 = \alpha\)

Draw a conclusion

Since the p-value is less than \(\alpha\), we reject the null hypothesis (\(H_0\)). There is statistically significant evidence to conclude that the distributions of the two populations are different at the 5% significance level.

Key concepts

Hypothesis Testing

In statistics, hypothesis testing is a systematic method for evaluating two competing claims or hypotheses about a population, based on sample data. The process begins by establishing a null hypothesis (\( H_0 \)), which represents a default statement of no effect or no difference, and an alternative hypothesis (\( H_A \( or \( H_1 \)), which is what we seek evidence for.

In the given exercise, the null hypothesis asserts that there is no difference in the distribution between two non-normal populations. The alternative hypothesis suggests that their distributions differ. After formulating the hypotheses, we calculate a test statistic from the sample data, which measures how compatible our data is with the null hypothesis.

To determine the result, we compare a calculated p-value to a predetermined significance level, \( \alpha \). If the p-value is less than \( \alpha \), we have enough evidence to reject the null hypothesis and accept the alternative. Conversely, if the p-value is greater, we fail to reject the null hypothesis, implying insufficient evidence to support the alternative.
In our exercise, using the Mann-Whitney U test, a nonparametric test statistic, we find a p-value that allows us to reject the null hypothesis at the 5% significance level, indicating a significant difference between the two population distributions.

P-value Calculation

The p-value, or probability value, is a crucial aspect of hypothesis testing which quantifies the strength of the evidence against the null hypothesis. It represents the probability of observing test results at least as extreme as the ones obtained, assuming the null hypothesis is true.

To calculate the p-value in nonparametric tests like the Mann-Whitney U test, we often use special tables or statistical software that accommodate the distributions of non-normal datasets. Once we have the test statistic, such as \( U_2 = 42 \) in our case, we look up this value in a Mann-Whitney table or input it into a software to obtain the p-value.

For this exercise, we found the p-value to be approximately 0.018, which we then compared with our alpha level of 0.05. Because our p-value is less than \( \alpha \), we conclude that our results are statistically significant, allowing us to reject the null hypothesis. Understanding p-values is essential not only for determining the outcome of a test but also for interpreting the result in context, such as understanding the potential for Type I errors (false positives).

Nonparametric Statistics

Nonparametric statistics provide methods to analyze data that do not require the data to fit a particular distribution, which is a common assumption for many traditional parametric tests. They are particularly useful for analyzing ordinal data or data that do not meet the normal distribution assumption, either due to small sample sizes or inherent non-normality.

The Mann-Whitney U test, utilized in this textbook exercise, is a classic example of a nonparametric test. It compares the distributions of two independent samples to assess whether there is a difference in their population distributions. Instead of means or variances, the Mann-Whitney U test analyzes ranks of the data, offering a robust analysis without the need for the data to conform to the normal distribution.

Applying nonparametric methods like the Mann-Whitney U test involves computing values such as \( U_1 \) and \( U_2 \) based on sample sizes and ranks, followed by p-value calculations to determine the significance of the results. These methods are invaluable when dealing with real-world data that often deviate from the ideal conditions required for parametric tests.