Rank Correlation Use the ranks from Exercise 1 to find the val...

Question

Rank Correlation Use the ranks from Exercise 1 to find the value of the rank correlation coefficient. Also, use a 0.05 significance level and find the critical value of the rank correlation coefficient. What do you conclude about correlation?

Accepted Answer

All right. Hello, everyone. So this question says, to use the ranks from the data provided below to calculate the Spearman rank correlation coefficient. Then, using a 0.05 significance level, find the critical value and determine whether there is a statistically significant correlation between the two variables. So here we have a data set with 5 students labeled A through E listing their rank both in math and science. And first, I want to take a moment to list both the null and alternative hypothesis. Now, the null hypothesis would state that there is no correlation between the two sets of rankings. Which means that the correlation coefficient would be equal to zero. Now, the alternative hypothesis would state the opposite, right, that there is instead a correlation between both sets of rankings. So in the null hypothesis, where R is equal to 0, in the alternative hypothesis, R is not equal to 0. And from here we proceed to calculating the Spearman rank correlation coefficient. So, with our data table that's already given to us, I'm going to. Use it as an extension to find more information. Right, so let's say that the ranking in math for each student is X and in science it's Y. The difference, lowercase d is equal to X subtracted by Y. So let's take a moment to calculate the differences between them. Right, and going from student A to student E D. Equals the following, right, we have -1. Positive 2 -1, 1, and 1. So now that you found the values for D for each student, in the next column that I'm making here, you would then square it. So -1 squared is 1. 2 squared is 4. -1 squared would give you 1. And 1 squared would also give you 1. So now We can proceed to calculating the actual coefficient. And I had pre-written the formula here in advance just to save a little bit more time. So now I'm just bringing it up so you can see it. All right, so RS, which is the correlation coefficient, is equal to 1 subtracted by 6 multiplied by the sum of all D2d values, divided by N. Multiplied by n squared subtracted by 1. With n being the number of students. So very quickly, the sum of all these squared values is equal to 8. So R S is equal to 1, subtracted by 6, multiplied by 8. And divided by the sample size here, the number of students. Which is 5 multiplied by 5 squares subtracted by 1. And this ultimately equals 0.6. So now From here we would determine the critical value. So from Spearman's rank correlation table, right, N is equal to 5 because there's 5 students, and as mentioned before, alpha is equal to 0.05. And this is a two-tailed test, which means that the critical value is positive or negative, 0.900. So Here, our calculated value, as mentioned before, is 0.6. Because the absolute value of 0.6 is less than 0.900, we fail to reject the null hypothesis. So now this brings us to our final answer, which reads as follows. R S is equal to 0.6, and there is insufficient evidence to conclude a significant correlation between the two sets of ranks. And there you have it. So with that being said, thank you so very much for watching and I hope you found this helpful.

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