## Exercises - Spearman's Rank Sum Correlation Test

1. Dr. Kon Seison ranked ten of his calculus students from highest to lowest grade in the course. In order to compare student performance in the course with their math SAT scores, he compiled the following table. $$\begin{array}{l|cccccccccc} \textrm{Student} & Alan & Bart & Beth & Bob & Fred & Gary & Hugh & Ivy & Lenny & Uma\\ \hline\hline \textrm{Math SAT} & 585 & 582 & 565 & 577 & 573 & 581 & 589 & 559 & 569 & 599 \\ \textrm{Ranking} & \textrm{10th} & \textrm{2nd} & \textrm{6th} & \textrm{4th} & \textrm{7th} & \textrm{8th} & \textrm{9th} & \textrm{5th} & \textrm{3rd} & \textrm{1st} \end{array}$$ Is there any relationship between students' ranking in the course and their SAT test scores at 0.05?

Which test is more appropriate to use: the parametric correlation test or the Spearman's rank correlation test? Why?

Use a non-parametric test because rankings are ordinal.

Null hypothesis: $\rho_S=0$

Test statistic: $r_S=.055\ (\Sigma d^2=156)$

Critical value: $\pm .648$

Fail to reject the null hypothesis. Test statistic is not in the rejection region.

There is not enough evidence to support the claim that there is a relationship between students' ranking in the course and their SAT test scores.

2. A consumer group compared ratings of toaster ovens to price for a random sample of ovens. At significance level 0.05, is there a correlation between ratings and prices? Explain why we should use a non-parametric test. $$\begin{array}{r|ccccccc} \hbox{Model} &A&B&C&D&E&F&G\\\hline \hbox{Rating(1-10)} & 3 & 4 & 6 & 5 & 7 & 10 & 9\\ \hbox{Price(\)}&25&49&30&59&55&35&70\\ \end{array}$$

Use a non-parametric test because ratings are ordinal.

Null hypothesis: $\rho_S=0$

Test statistic: $r_S=.39285\ (\Sigma d^2=34)$

Critical value: $\pm .786$

Fail to reject the null hypothesis. Test statistic is not in the rejection region.

There is not enough evidence to support the claim that there is a correlation between ratings and prices.