Evaluating the correlation of data values that produce box patterns

Intro

Howdy!  I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help.  Today we're going to learn how to evaluate the correlation of data values that produce box patterns.  Here's our problem statement: Refer to the accompanying scatterplot.  The four points in the lower left corner are measurements from women, and the four points in the upper right corner are from men.  Complete Parts A through E below.

Part A

OK, Part A says, “Examine the pattern of the four points in the lower left corner (from women) only and subjectively determine whether there appears to be a correlation between x and y for women.  Choose the correct answer below.”

We look here at these four points that are referenced here.  These are the ones for women.  We see that they form a box pattern.  The correlation we're asked to find is a linear correlation; if you look at all of your answer options, they are all asking about a linear correlation.  Well, the points here are not forming a straight line.  Therefore, we can subjectively say there's not likely to be a linear correlation.  So there does not appear to be a linear correlation because the points do not form a line.  Excellent!

Part B

Part B says, “Examine the pattern of the four points in the upper right corner (from men) only and subjectively determine whether there appears to be a correlation between x and y for men.  Choose the correct answer below.”

Well, we can see that these data points up here forming a same pattern as for the women.  So we're going to conclude the same answer that we concluded for the women.  There does not appear to be a linear correlation because the points do not form a line.  Good job!

Part C

Part C says, “Find the linear correlation coefficient using only the four points in the lower left corner for women.  Will the four points in the upper right corner for men have the same linear correlation coefficient?”

OK, the first part here is to find the linear correlation coefficient for the data points corresponding to the women.  That's these four data points here in the lower part of the graph.  To find the linear correlation coefficient, I'm going to use StatCrunch.  So here I have StatCrunch open.  Notice there's no icon in this problem statement that allows me to dump the individual data points into StatCrunch.  That means I'm going to have to transfer that information into StatCrunch by hand.

I'm going to call this first column X, and I'm going to call this second column Y.  And we can blow this graph up a bit so we can get a better look at the ordered pairs for our data points.  So this first one here looks to be (3, 1), so here in StatCrunch I'll put 3 1.  This is (4, 1), and this looks to be (3, 2), and this one looks to be (4, 2).

OK, so now I've got those four data points in StatCrunch.  Now I come up to Stat –> Regression –> Simple Linear.  I’m going to tell StatCrunch were to find my x- and y-values.  And then the other defaults are fine for our purpose, so I'm going to press Compute!, and here comes my results window.  Notice our correlation coefficient is zero, so that's what I'm going to put here in my answer field.  Excellent!

“Do the four points in the upper right corner have the same correlation coefficient?”  Well, those four points are forming the same pattern as the four on the bottom.  So it's good to conclude that, yes, they're going to form the same pattern.  Therefore they have the same correlation coefficient.  Nice work!

Part D

Now Part D says, “Find the value of the linear correlation coefficient using all eight points.  What does that value suggest about the relationship between x and y?  Use alpha equals 0.05.”  OK, to do this I'm going to have to put in the remaining four points from the men into my data set.  So back here in StatCrunch, I'm going to put the four points into StatCrunch.  This first one looks to be (9, 9), which means this next one is (10, 9).  Then we have (9, 10), and this last point is (10, 10).

Now to get back to my options window, I simply go to my results window, press the Options button, select Edit, and here I'm in my options window again.  I've already told StatCrunch where to find my x- and y- values, so all I need to do is repeat the calculation by pressing Compute!, and now I have a new correlation coefficient.  You can see right here I'm asked to round to three decimal places.  Well done!

Now we're asked, “Using alpha equals 5%, what does R suggest about the relationship between x and y?”  Well, to help us understand the answer to this question, I prepared a brief PowerPoint presentation that helps us run through what we need to know in order to evaluate whether or not the correlation is strong enough to be considered useful.  We need to evaluate the critical R value.  That comes from a table of critical R values, so here I have a sampling of such a table.  There's actually one of these tables, I believe, in the insert to your textbook.

Be that as it may, the way that we use the table is first we have to identify the number of data points, or pairs of data, that we have in our data set.  We've got eight dots here in our scatter plot, so we have eight data points.  So the first thing we do is identify the total number of data points in our data set.  And then we read the value to the right of that based on our alpha level.  So our alpha level from the problem listed here is five percent, so that means we're going to have a critical R value of 0.707.

So our critical R value is 0.707.  We need to compare that with the R value obtained from the actual data.  In this case, that's going to be 0.979.  When we compare these two values, we find that our R value is greater than our critical R value, and therefore because the actual R value is greater than the critical R value, we've exceeded that threshold that we need for making a conclusion of correlation.  And therefore we can reject the null hypothesis that the correlation coefficient is zero and conclude that there is sufficient evidence to support a claim of linear correlation.  That's the answer I'm going to select here from my different options.  Well done!

And now Part 2 asks, “Based on the preceding results, what can be concluded?  Should the data from women and the data for men be considered together, or do they appear to represent two different and distinct populations that should be analyzed separately?”  To answer these questions, I'm going to go back here to my scatter plot and look to see where my data lie.  All of the women — the data points for all the women — are congregated down here.  However, all the men — the data points for the men — are congregated in a separate grouping up here.  There's no mixing of the data points.  There's no meshing of them together there on the graph.  So they're occupying distinct areas or regions of the graph.  This suggests that they are two distinct populations and should therefore be considered separately.  I'm going to select that answer from my options here.  Nice work!

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