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. An educational researcher believes that the median number of faculty for proprietary (for-profit) colleges and universities is $150 .$ The data provided list the number of faculty at a randomly selected number of proprietary colleges and universities. At the 0.05 level of significance, is there sufficient evidence to reject his claim? $$ \begin{array}{rrrrrrrrrr} 372 & 111 & 165 & 95 & 191 & 83 & 136 & 149 & 37 & 119 \\ 142 & 136 & 137 & 171 & 122 & 133 & 133 & 342 & 126 & 64 \\ 61 & 100 & 225 & 127 & 92 & 140 & 140 & 75 & 108 & 96 \\ 138 & 318 & 179 & 243 & 109 & & & & & \end{array} $$

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.
An educational researcher believes that the median number of faculty for proprietary (for-profit) colleges and universities is $150 .$ The data provided list the number of faculty at a randomly selected number of proprietary colleges and universities. At the 0.05 level of significance, is there sufficient evidence to reject his claim?
$$
\begin{array}{rrrrrrrrrr}
372 & 111 & 165 & 95 & 191 & 83 & 136 & 149 & 37 & 119 \\
142 & 136 & 137 & 171 & 122 & 133 & 133 & 342 & 126 & 64 \\
61 & 100 & 225 & 127 & 92 & 140 & 140 & 75 & 108 & 96 \\
138 & 318 & 179 & 243 & 109 & & & & &
\end{array}
$$

Key Concepts

Hypothesis Formulation

This concept involves setting up two competing statements: the null hypothesis (H0) and the alternative hypothesis (Ha). The null hypothesis typically represents a statement of no effect or a status quo (for example, that the median is equal to a specific value), while the alternative hypothesis represents the claim or effect the researcher is testing for (for instance, that the median differs from that value). In hypothesis testing, the claim made by the researcher is identified as either H0 or Ha based on the context and the direction of the test.

Significance Level

The significance level, denoted by alpha (?), is a threshold for deciding whether the observed result is statistically significant. It represents the probability of making a Type I error, which is rejecting a true null hypothesis. In many studies, including this context, a common significance level used is 0.05. This value is used to set the boundaries (critical values) for the decision rule in hypothesis testing.

Critical Value

Critical values are the cutoff points that define the rejection region for the test statistic under the assumption that the null hypothesis is true. They are determined from the relevant probability distribution (such as the normal distribution, t-distribution, or a nonparametric distribution, depending on the test) corresponding to the chosen significance level. These values are used in the decision process to determine whether the observed test statistic is extreme enough to reject the null hypothesis.

Test Statistic Calculation

The test statistic is a numerical value computed from the sample data that measures the degree of departure from what is expected under the null hypothesis. Depending on the type of test being conducted, the computation method can vary. For a test involving a median, a nonparametric method such as the sign test might be used. This calculated statistic is compared against the critical value(s) to decide the outcome of the hypothesis test.

Decision and Conclusion

After computing the test statistic and identifying the critical value(s), the decision rule is applied: if the test statistic falls within the rejection region defined by the critical value(s), the null hypothesis is rejected; otherwise, it is not rejected. The final step involves summarizing these results in context, meaning that the conclusion should state whether there is sufficient evidence to support or refute the researcher's claim regarding the median, and discussing this in terms of practical significance.

Transcript

Please give Ace some feedback