Question

A political scientist is studying the relationship between the voter image of a conservative political candidate and the distance (in kilometers) between the residences of the voter and the candidate. Each of 12 voters rated the candidate on a scale of \(1-20\). a. Calculate Spearman's rank correlation coefficient \(r_{s}\) b. Do these data provide sufficient evidence to indicate a negative rank correlation between rating and distance?

Short answer

Calculate Spearman's rank correlation coefficient and provide your determination.

Step-by-step solution

Create a table with distance and ratings sorted in ascending order of distance. #tag_step2# Calculate the rank difference

Calculate the differences between the ranks of distances and ratings (\(D_i = R_{distance_i} - R_{rating_i}\)) for each voter. #tag_step3# Square the rank differences

Square the rank differences for each voter (\(D_i^2)\). #tag_step4# Calculate the sum of squared rank differences

Add up all the squared rank differences from step 3 (\(\sum D_i^2)\). #tag_step5# Calculate Spearman's rank correlation coefficient

Use the following formula for \(r_{s}\): \(r_{s} = 1 - \frac{6\sum D_i^2}{n(n^2 - 1)}\), where n is the number of observations (12 voters). #tag_step6# Determine the evidence for negative rank correlation

Assess whether the calculated \(r_{s}\) value provides enough evidence to indicate a negative rank correlation between rating and distance.

Key concepts

Rank Correlation

Rank correlation measures the relationship between the rankings of two variables or data sets. When we talk about Spearman's rank correlation coefficient, often denoted by \(r_s\), we're focusing on a non-parametric measure that assesses how well a relationship between two variables can be described using a monotonic function. This means that as one variable increases, the other either tends to increase or decrease consistently.

For example, in the given exercise, the relationship between voters' ratings of a political candidate and their distance from the candidate's residence is being analyzed. If closer proximity to the candidate's residence is consistently associated with higher ratings, then a strong positive rank correlation would exist. Conversely, if greater distance is associated with higher ratings, this would constitute a negative rank correlation.

To calculate Spearman's \(r_s\), you first rank each variable, calculate the differences in ranks for each pair of observations (\(D_i\)), square these differences (\(D_i^2\)), and finally apply them in the Spearman's formula. Simplifying statistical evidence into manageable steps like this helps make data analysis more approachable for students.

Statistical Evidence

Statistical evidence refers to the use of statistical data to determine the likelihood of a certain hypothesis being true or false. In the case of Spearman's rank correlation coefficient, statistical evidence can affirm or reject the presence of a reliable association between two ranked variables.

When evaluating statistical evidence, it's crucial to consider the significance level, which helps determine the reliability of the results. In the context of this exercise, after calculating \(r_s\), we would compare it against a critical value from a reference table that corresponds to our desired level of significance (often 0.05 for a 5% significance level). If the absolute value of \(r_s\) is greater than the critical value, we have enough evidence to suggest a significant rank correlation.

Furthermore, the direction of \(r_s\) indicates the nature of the correlation; if \(r_s \) is negative and significant, we confirm a negative rank correlation, which would match the scenario proposed in the exercise - whether a greater distance implies a lower rating of the political candidate.

Data Analysis

Data analysis involves examining, transforming, and arranging data to discover useful information, draw conclusions, and support decision-making. In the context of Spearman's rank correlation, data analysis would include organizing data (such as by creating a table with ranked distances and ratings), performing calculations step by step, and interpreting results.

In the example provided, data analysis allowed the political scientist to quantify and understand the relationship between voter perceptions and their geographic proximity to the candidate. By systematically following the steps for calculating Spearman's \(r_s\), and then inferring the statistical evidence of this coefficient, the analyst can conclude if the initial hypothesis of negative rank correlation holds true.

Effective data analysis also includes critically assessing the data and the methodology, like checking for outliers or considering if the sample size is adequate. By fostering a comprehensive understanding of these analytical steps, we empower students to confidently apply these principles in various real-world scenarios.