Question

A high school principal formed a review board consisting of five teachers who were asked to interview 12 applicants for a vacant teaching position and rank them in order of merit. Seven of the applicants held college degrees but had limited teaching experience. Of the remaining five applicants, all had college degrees and substantial experience. The review board's rankings are given in the table. $$ \begin{array}{cc} \hline \text { Limited Experience } & \text { Substantial Experience } \\ \hline 4 & 1 \\ 6 & 2 \\ 7 & 3 \\ 9 & 5 \\ 10 & 8 \\ 11 & \\ 12 & \\ \hline \end{array} $$ Do these rankings indicate that the review board considers experience a prime factor in the selection of the best candidates? Test using \(\alpha=.05 .\)

Short answer

To determine if experience is a prime factor in the selection of the best candidates, we used the Wilcoxon Rank-Sum test. We computed the rank sums for each group, the expected rank sum, the test statistic, and found the critical value. Since the test statistic (19) was smaller than the critical value (23), we rejected the null hypothesis. This indicates that there is a significant difference between the rankings of applicants with limited experience and those with substantial experience, suggesting that experience is a prime factor in the selection of the best candidates.

Step-by-step solution

Compute the rank sums for each group

We will start by calculating the rank sums of the groups with limited experience (Group A) and with substantial experience (Group B). Group A: \(4 + 6 + 7 + 9 + 10 + 11 + 12 = 59\) Group B: \(1 + 2 + 3 + 5 + 8 = 19\)

Compute the expected rank sum for each group

We will calculate the expected rank sum for groups A and B, given the total number of applicants in each group: $$ E_{A} = \frac{n_{A}(n_{A} + n_{B} + 1)}{2} $$ $$ E_{B} = \frac{n_{B}(n_{A} + n_{B} + 1)}{2} $$ where \(n_A\) is the number of applicants in group A and \(n_B\) is the number of applicants in group B. For group A: $$ E_{A} = \frac{7(7 + 5 + 1)}{2} = \frac{7 \cdot 13}{2} = 45.5 $$ For group B: $$ E_{B} = \frac{5(7 + 5 + 1)}{2} = \frac{5 \cdot 13}{2} = 32.5 $$

Compute the test statistic

Now we need to compute the Wilcoxon test statistic \(W\) using the rank sums we calculated in step 1. We will use the smaller of the two rank sums, \(W = \min(R_{A}, R_{B})\). $$ W = \min(59, 19) = 19 $$

Determine critical value and make a decision

We will use a Wilcoxon critical value table (found online or in a statistics textbook) to find the critical value for our problem. For a total number of 12 applicants, \(n_A = 7\) and \(n_B = 5\), and a significance level of \(\alpha = 0.05\), the critical value is 23. Since our test statistic (\(W = 19\)) is smaller than the critical value (23), we reject the null hypothesis. This indicates that there is a significant difference between the rankings of the two groups of applicants, suggesting that the review board considers experience a prime factor in the selection of the best candidates.

Key concepts

Statistical Hypothesis Testing

In the pursuit of understanding patterns and relationships within data, statistical hypothesis testing stands as a crucial procedure in statistics. It provides a framework for decision-making by establishing a method to determine whether observed data supports a specific claim or theory. To begin with, a null hypothesis (\( H_0 \) is posited as a default statement that there's no effect or no difference. On the contrary, an alternative hypothesis (\( H_1 \) challenges this by suggesting there is an effect or a difference.

When applying this to educational assessments, as in the exercise, the null hypothesis could propose that teachers' academic evaluations are independent of whether candidates have limited or substantial teaching experience. Conversely, the alternative hypothesis suggests that these evaluations depend significantly on the teaching experience. The Wilcoxon rank-sum test, a non-parametric alternative to the t-test, is then utilized to compare the ranks of two independent samples. If the test statistic calculated falls within a critical range or beyond a critical value, the null hypothesis can be rejected in favor of the alternative hypothesis. The calculated p-value helps determine this by comparing it to the predefined significance level (\( \( alpha \( \))), a threshold for rejecting the null hypothesis. If the p-value is less than this significance level, there's enough evidence to suggest the alternative hypothesis is valid.

Understanding these fundamental concepts is paramount, as they form the bedrock of both statistical analysis and evidence-based decision-making in various domains, including education.

Non-Parametric Statistics

Non-parametric statistics offer a set of tools for analyzing data that do not assume a specific distribution for the population from which the data are derived. This characteristic is particularly useful when dealing with data that are ordinal (like ranks), nominal, or when the sample size is too small to validate the assumptions of parametric tests. The Wilcoxon rank-sum test, utilized in the exercise, is a prime example of a non-parametric test.

Non-parametric tests trade off some of the statistical power of their parametric counterparts for fewer assumptions and increased versatility. They rank the data points and then investigate the differences between these rankings to draw conclusions. For instance, the Wilcoxon rank-sum test used in the educational evaluation method compares two independent samples to assess whether their population mean ranks differ. It's ideal for situations like the teacher selection process in the exercise, where the data is ordinal, and the sample size may not justify the assumptions of a parametric test.

The integral idea behind non-parametric tests is that they provide a robust way to achieve reliable results without relying heavily on the nature of the data distribution. They are indispensable tools in statistical analysis for research fields where the applicability of parametric tests is limited.

Educational Evaluation Methods

Educational evaluation methods encompass a wide array of techniques and tools used to assess various aspects of education. These include the evaluation of teaching methods, instructional materials, student learning, and in our exercise, the assessment of potential educators. Evaluations can be formative, aiding in the ongoing improvement of the educational process, or summative, culminating in a final judgment about the effectiveness of an educational program or candidate.

In the provided exercise, the review board is faced with the task of evaluating teacher candidates, a summative evaluation situation. The board used ranking as a method to manifest their assessment of each candidate's merit, which is a classic application of ordinal data. To analyze such data, the educational evaluators employed the Wilcoxon rank-sum test, which is particularly suited for ordinal data evaluation. This test allowed them to statistically determine whether the ranks given to candidates with limited experience significantly differed from those with substantial experience.

Improving the Exercise

When addressing such evaluations, it's important to clarify the objectives, provide transparent criteria, and ensure that the chosen methods align with evaluation goals. The Wilcoxon rank-sum test aids in these objectives by providing a statistical basis for the comparison between groups. To enhance students' grasp, explaining the rationale behind the test, its constraints, and the interpretation of its results extends the educational value far beyond the exercise itself, fostering a deeper understanding and critical thinking skills.