a From the following statistics, test (with α=0.05 ) to determine whether the location of popula

step 1 We want to conduct the following hypothesis test at alpha equals 0 .05 signals. We want to test

A From the following statistics, test (with α=0.05 ) to...

a From the following statistics, test (with $\alpha=0.05$ ) to determine whether the location of population 1 is to the right of the location of population 2. $$ \begin{array}{ll} \hline T_1=1205 & n_1=30 \\ T_2=1280 & n_2=40 \\ \hline \end{array} $$ b Repeat part (a) with $T_1=1065$. c Discuss the effect on the test statistic and $p$-value of decreasing $T_1$ to 1065 .

a From the following statistics, test (with $\alpha=0.05$ ) to determine whether the location of population 1 is to the right of the location of population 2.
$$
\begin{array}{ll}
\hline T_1=1205 & n_1=30 \\
T_2=1280 & n_2=40 \\
\hline
\end{array}
$$
b Repeat part (a) with $T_1=1065$.
c Discuss the effect on the test statistic and $p$-value of decreasing $T_1$ to 1065 .

Key Concepts

Effect of Test Statistic Variation

Modifying the test statistic, such as by decreasing its value, directly influences the outcome of the hypothesis test. A reduction in the test statistic that compares against a critical value or expected distribution under the null hypothesis can result in a larger p-value, thereby reducing evidence against the null hypothesis. Essentially, changes in the test statistic—when compared to its distribution under the null hypothesis—determine the strength of evidence for one hypothesis over the other.

Test Statistic and p-value

The test statistic in nonparametric tests, like the Wilcoxon Rank-Sum test, is computed by summing the ranks of the observations for one sample and comparing this sum to its expected value under the null hypothesis. The p-value measures the probability of observing a test statistic as extreme as, or more extreme than, the one calculated, under the null hypothesis. A small p-value suggests that the observed data are unlikely under the null hypothesis, leading to its rejection at a predefined significance level (e.g., ? = 0.05).

Nonparametric Hypothesis Testing

Nonparametric hypothesis testing is used when the assumptions required for parametric tests (such as normality of the data) are in doubt or do not hold. These tests do not assume a specific probability distribution for the data and are often based on ranks rather than raw values, making them suitable for comparing location differences (central tendencies) between populations.

Wilcoxon Rank-Sum Test

The Wilcoxon Rank-Sum test, also known as the Mann-Whitney U test, is a nonparametric test used to compare the central location of two independent samples. It assesses whether one population tends to yield larger values than the other by ranking all the observations and evaluating the sum of ranks for each sample. This test is particularly useful when comparing median differences without assuming a normal distribution.

One-Sided Testing

One-sided testing involves setting up an alternative hypothesis that specifies a direction of the effect, such as testing whether the location of one population is greater than the other. In such tests, the rejection region is allocated to one tail of the sampling distribution, reflecting the hypothesis that a parameter (e.g., a median or location measure) lies strictly to one side of a reference value.

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