Question

Perform these steps. a. Find the Spearman rank correlation coefficient. b. State the hypotheses. c. Find the critical value. Use \(\alpha=0.05\). d. Make the decision. e. Summarize the results. Use the traditional method of hypothesis testing unless otherwise specified. Cyber School Enrollments Shown are the numbers of students enrolled in cyber school for five randomly selected school districts and the per-pupil costs for the cyber school education. At \(\alpha=0.10\), is there a relationship between the two variables? How might this information be useful to school administrators? $$ \begin{array}{l|ccccc} \text { Number of students } & 10 & 6 & 17 & 8 & 11 \\ \hline \text { Per-pupil cost } & 7200 & 9393 & 7385 & 4500 & 8203 \end{array} $$

Short answer

Reject the null hypothesis if the calculated \( r_s \) exceeds the critical value, indicating a significant relationship between variables.

Step-by-step solution

Assign Ranks to Each Variable

To find the Spearman rank correlation coefficient, first, assign ranks to each of the datasets. Rank both the number of students and the per-pupil costs. For instance, for the number of students: 6, 8, 10, 11, 17 would be ranked as 1, 2, 3, 4, 5 respectively. Similarly, rank the per-pupil costs.

Calculate Rank Differences

Next, calculate the differences between the ranks of each corresponding value pair (students, costs). Denote these differences as \( d_i \).

Compute Squared Rank Differences

Square each of the rank differences obtained in the previous step to get \( d_i^2 \).

Sum Up the Squared Differences

Find the sum of all squared rank differences, \( \sum d_i^2 \).

Calculate the Spearman Rank Correlation Coefficient

Use the formula for Spearman's rank correlation coefficient: \( r_s = 1 - \frac{6 \sum d_i^2}{n(n^2-1)} \), where \( n \) is the number of paired ranks.

State the Hypotheses

Set up the hypotheses: - Null Hypothesis \( H_0 \): There is no relationship between variables (\( \rho = 0 \)).- Alternative Hypothesis \( H_1 \): There is a relationship between variables (\( \rho eq 0 \)).

Find the Critical Value

Using the Spearman rank correlation critical value table, find the critical value for \( \alpha = 0.05 \) with \( n-2 = 3 \) degrees of freedom.

Make the Decision

Compare the calculated \( r_s \) with the critical value. If \( |r_s| \) is greater than the critical value, reject the null hypothesis.

Summarize the Results

Conclude based on the decision made in the previous step. If the null hypothesis is rejected, it indicates there is a significant relationship between the number of students and per-pupil costs. This information can help school administrators understand and possibly optimize resource allocation.

Key concepts

Hypothesis Testing

Hypothesis testing is a fundamental concept in statistics used to determine whether there is enough evidence to suggest a particular hypothesis about a data set is true. It involves a systematic method of evaluating statements and making decisions based on sample data.

In our context, hypothesis testing allows us to determine if there is a significant relationship between the number of students enrolled in cyber school and the per-pupil costs of the education. To conduct hypothesis testing, we follow these key steps:

• State the null hypothesis (H_0), which in this scenario assumes there is no relationship (\rho = 0).
• State the alternative hypothesis (H_1), indicating that there is a relationship between the two variables (\rho eq 0).
• Select a significance level, commonly denoted as \(\alpha\). For example, in this situation, \(\alpha = 0.05\).
• Calculate the test statistic using the Spearman rank correlation coefficient.
• Compare the calculated value to the critical value from the Spearman's rank correlation table.
• Make a decision to accept or reject the null hypothesis.

Hypothesis testing provides a structured approach to draw meaningful conclusions from data, which is essential for making informed decisions.

Cyber School Enrollments

Cyber school enrollments refer to the number of students who are participating in online schooling environments. As educational technology advances, understanding the dynamics and factors influencing enrollments in cyber schools becomes important.

In this study, we are examining the connection between the number of students enrolled and the costs per pupil. For school administrators, this analysis could shed light on how costs vary with the number of students enrolled. For example:

• Budgeting and Resource Allocation: Understanding this relationship might help administrators allocate resources more efficiently.
• Cost Management: Insights into cost behavior can lead to better strategies to manage or reduce unnecessary expenses.
• Tailored Educational Strategies: Information about costs versus enrollments can lead to more personalized educational approaches.

The ability to predict how enrollment numbers influence or correlate with costs can be extremely beneficial in strategic planning for educational institutions.

Rank Differences

To delve deeper into the calculation of Spearman's rank correlation coefficient, it is essential to understand the importance of rank differences. When we rank data points, we essentially convert them into a form that allows comparison without assuming a linear relationship.

Here's how the process works for our exercise involving cyber school enrollments:

• Assign ranks to both the number of students and their corresponding per-pupil costs.
• Calculate the difference between these ranks for each data pair, known as the rank differences and denote these as \( d_i \).
• Square these differences to eliminate any negative values, giving us \( d_i^2 \).
• The sum of these squared differences, \( \sum d_i^2 \), is then used in the formula to find Spearman's rank correlation coefficient.

Rank differences help in assessing how closely two variables are related in their relative ordering, providing evidence for any potential correlations.

Critical Value for Spearman's Rank

The critical value for Spearman's rank correlation is a threshold that helps in hypothesis testing to determine the significance of the computed rank correlation coefficient ( r_s). It's essential for assessing whether the observed correlations in the sample data are likely to reflect a true relationship in the population.

The critical value depends on two main factors:

• The chosen level of significance, \(\alpha\), which signifies the probability of rejecting the null hypothesis when it is actually true. Common values are 0.05 or 0.10.
• The degrees of freedom, which in the context of rank correlation, are calculated as \(n - 2\), where \(n\) is the number of data pairs.

Once \(r_s\) is computed, it should be compared to the critical value. If the absolute value of \(r_s\) is greater than the critical value, the null hypothesis is rejected, indicating a significant correlation.

This part of hypothesis testing is crucial because it determines the robustness of our finding, ensuring that the correlation is not just due to random chance.