For Exercises 5 through 20, perform these steps. a. State...

For Exercises 5 through $20,$ perform these steps. a. State the hypotheses and identify the claim. b. Find the critical value(s). c. Compute the test value. d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. The U.S. median annual income for men in 2014 (in constant dollars) was $\$ 35,642 .$ A random sample of recent male college graduates indicated the following incomes. At the 0.05 level of significance, test the claim that the median is more than $\$ 35,642$ $$ \begin{array}{lllll} 35,000 & 37,682 & 39,800 & 32,500 & 30,000 \\ 41,050 & 36,198 & 31,500 & 29,650 & 35,800 \\ 34,500 & 38,850 & 39,750 & & \end{array} $$

For Exercises 5 through $20,$ perform these steps.
a. State the hypotheses and identify the claim.
b. Find the critical value(s).
c. Compute the test value.
d. Make the decision.
e. Summarize the results.
Use the traditional method of hypothesis testing unless otherwise specified.
The U.S. median annual income for men in 2014 (in constant dollars) was $\$ 35,642 .$ A random sample of recent male college graduates indicated the following incomes. At the 0.05 level of significance, test the claim that the median is more than $\$ 35,642$
$$
\begin{array}{lllll}
35,000 & 37,682 & 39,800 & 32,500 & 30,000 \\
41,050 & 36,198 & 31,500 & 29,650 & 35,800 \\
34,500 & 38,850 & 39,750 & &
\end{array}
$$

Key Concepts

Nonparametric Methods (Sign Test)

When testing for a median, especially without assuming a normal distribution, nonparametric methods such as the sign test are used. This approach involves examining the signs of the differences between observed values and the test median, thereby providing an alternative to parametric tests when the sample size is small or distributional assumptions are in doubt.

Critical Values and Decision Rule

Critical values are specific points on the distribution of the test statistic under the null hypothesis that divide the acceptance region from the rejection region. Establishing these values using the significance level allows the researcher to determine a decision rule—if the test statistic exceeds the critical value (in a one-tailed test), then the null hypothesis is rejected.

Test Statistic

The test statistic is a calculated value derived from the sample data that is used to decide whether to reject the null hypothesis. It quantifies how far the sample statistic is from the parameter value stated under the null hypothesis, and it is subsequently compared to critical values or used to compute a p-value.

Null and Alternative Hypotheses

In hypothesis testing, you begin by stating the null hypothesis (H0), which typically reflects a status quo or a statement of no effect or difference, and the alternative hypothesis (Ha), which represents the claim you wish to validate. In the context of median testing, H0 might state that the median is at most a certain value, while Ha indicates that the median is greater than that value.

Significance Level

The significance level, denoted by alpha (?), is a predetermined threshold that defines the probability of rejecting the null hypothesis when it is actually true (Type I error). It plays a crucial role in setting the boundary for establishing the rejection region in the test (e.g., ? = 0.05).

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