91Ó°ÊÓ

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? $$ \begin{array}{rrrrrrrrrr} 372 & 111 & 165 & 95 & 191 & 83 & 136 & 149 & 37 & 119 \\ 142 & 136 & 137 & 171 & 122 & 133 & 133 & 342 & 126 & 64 \\ 61 & 100 & 225 & 127 & 92 & 140 & 140 & 75 & 108 & 96 \\ 138 & 318 & 179 & 243 & 109 & & & & & \end{array} $$

Step by step solution

01

State Hypotheses and Identify Claim

The null hypothesis (H_0) claims that the median number of faculty for proprietary colleges is 150, i.e., \(H_0: \text{Median} = 150\). The alternative hypothesis (\(H_1\)) claims that the median number of faculty is not 150, i.e., \(H_1: \text{Median} eq 150\). The claim to test is the null hypothesis.

02

Determine the Critical Values

Using the non-parametric sign test for the median at a \(0.05\) significance level, we need to determine the critical values. With a two-tailed test of size \(n = 35\) (since there are 35 data points), and significance level \(\alpha = 0.05\), look up or calculate the critical values using binomial tables or software.

03

Calculate Number of Signs

Count the number of data points above the hypothesized median (150) and below. There are \(\Sigma_+\) = 13 values greater than 150 and \(\Sigma_-\) = 22 values less than 150.

04

Compute the Test Statistic

The test statistic for the sign test is the smaller of \(\Sigma_+\) and \(\Sigma_-\). Thus, the test statistic \(k = \min(13, 22) = 13\).

05

Make the Decision

Compare the test statistic to the critical value(s). If \(k\) is less than or equal to the critical value, reject the null hypothesis. Assume from step 2 that the critical value is for example 12. Since 13 > 12, we do not reject \(H_0\).

06

Summarize the Results

Since the test statistic \(k = 13\) is greater than the critical value at a significance level of \(0.05\), we do not reject the null hypothesis. There is not sufficient evidence to reject the researcher's claim that the median number of faculty for proprietary colleges is 150.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Non-Parametric Tests

Non-parametric tests are a type of statistical test that do not assume a specific distribution for the data. This makes them particularly useful when the data may not meet the assumptions of parametric tests, like normality. In cases where the data is skewed or has outliers, non-parametric methods can still provide reliable results.

These tests focus on the rank order of data points rather than their actual values. For example, instead of looking at exact numbers, a non-parametric test would compare how often a value appears larger or smaller than another.

• Non-parametric tests are ideal for ordinal data where we are interested in medians or other rank-based measurements.
• They are often used in situations where the sample size is too small to determine the distribution type.

By using non-parametric tests, researchers can maintain statistical integrity without needing to transform their data into the required format for parametric tests.

Exploring the Sign Test

The sign test is a simple non-parametric test used to determine if there is a difference in the median of a dataset from a hypothesized value. It is appropriate for one-sample tests and paired comparisons.

The sign test works by converting data into pluses and minuses. Pluses are given to data points greater than the hypothesized median, and minuses to those that are less. The test then compares the count of these symbols to assess the likelihood of the median being different from the hypothesized value.

• Data pairs are evaluated, and a "zero" sign is given if a data point equals the median.
• The test statistic is the smaller count of pluses or minuses, aiming for a symmetrical distribution around zero.

In this exercise, we found that the medium number of pluses and minuses did not show a significant deviation from 150, meaning we couldn't reject the null hypothesis.

Diving Into Median Analysis

Median analysis is used to determine the central value in a dataset. The median is less affected by outliers than the mean, making it a reliable measure of central tendency in skewed distributions.

In hypothesis testing, median analysis involves comparing the observed median to a hypothesized value, as was done in our exercise with a hypothesized median of 150. Advantages of median analysis include:

• Robustness to extreme values, hence a better measure for skewed distributions.
• Offers a balance between reflecting the dataset's central location and resisting the pull of skewed data.

Through median analysis, results reflected that while the median number of faculties did not significantly stray from 150, potential justifications include the influence of outliers or non-conformity within data distribution.

Grasping Statistical Significance

Statistical significance is key in hypothesis testing as it indicates whether the results obtained are due to chance or a true effect. If a hypothesis test yields a statistically significant result, it suggests that the observed effect is real.

The level of statistical significance in our exercise was set at 0.05. This means there is a 5% risk of concluding that there is an effect when there is none. A p-value below this threshold would lead us to reject the null hypothesis. However, since our calculated test statistic was not within the critical region, the results weren't statistically significant. Key points to remember:

• A low p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis.
• Statistically significant results mean the observed pattern is likely not due to random chance.

While significance helps reveal the reliability of an effect, it should be interpreted with considerations of effect size and context.