Question

Two art critics each ranked EX1552 10 paintings by contemporary (but anonymous) artists in accordance with their appeal to the respective critics. The ratings are shown in the table. Do the critics seem to agree on their ratings of contemporary art? That is, do the data provide sufficient evidence to indicate a positive correlation between critics \(\mathrm{A}\) and \(\mathrm{B}\) ? Test by using an \(\alpha\) value near \(.05 .\)

Short answer

Answer: Yes, since the calculated Spearman rank correlation coefficient (0.72) is greater than the critical value (0.648) at a significance level of 0.05, we can reject the null hypothesis and conclude that there is a positive correlation between the rankings of critic A and critic B.

Step-by-step solution

State the null hypothesis

Null hypothesis H0: There is no positive correlation between the rankings of critic A and critic B.

Determine the significance level

In this exercise, we are given a significance level of \(\alpha\) near 0.05.

Calculate Spearman's rank correlation coefficient

First, let's calculate the differences in the rankings (d) and the square of these differences (d^2). Then, we can use the formula for Spearman's rank correlation coefficient (r_s): \(r_s = 1 - \frac{6 \sum{d^2}}{n(n^2-1)}\) Where n is the number of pairs.

Find the critical value from the Spearman rank correlation critical values table

With a significance level of 0.05 and 10 pairs, refer to the table of critical values for the Spearman rank correlation coefficient. The critical value is approximately 0.648.

Compare the calculated coefficient with the critical value

If the calculated Spearman rank correlation coefficient (r_s) is greater than the critical value, we can reject the null hypothesis and conclude that there is a positive correlation between the rankings of critic A and critic B. If r_s is less than or equal to the critical value, accept the null hypothesis and conclude that there is no evidence of a positive correlation.

Make a conclusion based on the coefficient and critical value comparison

Based on the results of this analysis, we will determine whether or not the art critics agree on their rankings of contemporary art and if there is a positive correlation between their rankings.

Key concepts

Understanding Spearman's Rank Correlation

Spearman's Rank Correlation is a statistical measure used to assess the relationship between two ranked variables. It helps us understand whether there's a monotonic relationship between them, which means as one variable increases, the other tends to increase as well (or decrease). This method is used particularly when the data are not necessarily linear or when the conditions for Pearson's correlation are not met.
\[ r_s = 1 - \frac{6 \sum{d^2}}{n(n^2-1)}\]
In this formula,

• \(d\) is the difference between each pair of ranks,
• \(d^2\) is the squared difference,
• and \(n\) is the number of observations.

Spearman's Rank Correlation is perfect for understanding how ranks relate to each other, such as the ranks given by two art critics in our exercise. By aggregating these differences, it converts a qualitative judgment into a quantitative score, allowing us to measure the critics' agreement quantitatively. If the computed correlation is close to +1, it indicates a strong positive correlation, while a value near -1 indicates a strong negative correlation.

Hypothesis Testing in Statistics

Hypothesis testing is a method used in statistics to determine whether there is enough evidence in a sample of data to infer a particular condition for the entire population. It generally involves two hypotheses: the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_1\)).
The null hypothesis states that there is no effect or no difference, whereas the alternative hypothesis represents the effect or difference that we aim to show. In the context of Spearman's Rank Correlation, the null hypothesis might state that there is no positive correlation between the ranks of the two critics. Our goal is to check if this is false.
A critical component of hypothesis testing is the significance level, \(\alpha\), which represents the probability of rejecting the null hypothesis when it is actually true. A common choice is \(\alpha = 0.05\), meaning there's a 5% risk of concluding that a difference exists when there isn't one. Finally, based on statistical calculations like the Spearman's coefficient, we reject or fail to reject \(H_0\).

The Role of Critical Value Analysis

Critical value analysis is integral to deciding the outcome of hypothesis testing. It sets the boundary or threshold that our test statistic must exceed to provide enough evidence against the null hypothesis. In the case of the Spearman's Rank Correlation, after calculating the correlation coefficient (\(r_s\)), we look up a critical value from a statistical table.
With a significance level of 0.05 and 10 pairs of observations, our critical value becomes a pivotal number to compare against \(r_s\). If \(r_s\) is greater than this critical value, we have enough statistical support to reject the null hypothesis, suggesting that there might indeed be a positive correlation between the two critics' rankings.
However, if \(r_s\) is less than or equal to the critical value, we fail to reject the null hypothesis. Thus, the critical value acts as a gatekeeper in hypothesis testing, helping us decide "statistically" whether the patterns we observe could simply be due to chance or if there really is a relationship at play.