Clusters Refer to the Minitab-generated scatterplot. The four ...

Question

Clusters Refer to the Minitab-generated 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.

Scatterplot showing eight data points, with four in the lower left and four in the upper right, representing two distinct groups.

Find the value of the linear correlation coefficient using all eight points. What does that value suggest about the relationship between x and y?

Accepted Answer

Welcome back, everyone. By referring to the following scatter plot, we want to determine the linear correlation coefficient for the given set of points. What does its value indicate about the relationship between X and Y? A says the linear correlation coefficient R is -0.92 indicating a strong negative correlation. B says it's 0.92 indicating a strong positive correlation. C says it's 0 indicating no correlation, and the D says it's 0.5 indicating a moderate positive correlation. Now here How can we use our scatter plot to find a linear correlation coefficient for our set of points? What do we know about this coefficient? We recall, OK. That the pier correlation coefficient r is equal to the number of values n multiplied by the sum of X multiplied by y where X and Y are the X and Y coordinates of each point respectively minus the sum of X values. Multiplied by the sum of y values. All divided by the square root. Of n multiplied by the sum of X squared values minus the sum of x values squared multiplied by the square root of n multiplied by the sum of y squared values minus the sum of y values squared. So for us to find the value of the piercing correlation coefficient, then we'll need to find our pairs of points for the Xy values and then solve for a few things here. We need the sum of Xy values, the sum of X values, the sum of y values, and the sum of X2 values and the sum of Y squared values. Once we can get all of that, we can plug them into the formula to find the linear correlation coefficient. Now first of all, what are our pairs of points? Well, from our graph, notice that we have one tool, OK. We have 13. 22, OK. 23. And all of these are the points that are in this lower section of the graph. And then up here in the upper section, to the upper, upper right, we have 88. 89 OK, uh, 98. 99 And 910. So with these 9 pairs of points, we are going to go ahead and set up a table to help us find all the sums we need for the piers correlation coefficient. So let's get down here, OK? And we're gonna have our X values, Y values, or XY values. X squared and Y squared, OK. So let me just draw a line here. And create our table. OK. And now, let's put our pairs of points. So like we said earlier, we had 12. 13. 22 2388. 89. 98, OK, 99 and 910. So these are all our values. And I think as a matter of fact, I think we can make some space here and just bring this part of the table a little bit closer. OK. Let's just bring this over a little bit, yeah. And you know what? I feel like we could do the same for the others actually just to make some space as we're going along. Cuz no, all we're gonna do is we are going to put these values into our table. We're gonna compute the rest of the values that we need. So no, starting with, I think, I don't think we need this much for Y2. Starting with our points 1 and 2, OK. Then Xy would be 2, X2 is 1, and Y2 is 4. For 1 and 3, XY is 3, X2 is 1, and Y2 is 9. For 2 and 2, XY is 4, X2 is 4, Y2 is 4, for 2 and 3. XY is 6, X2 is 4, Y2 is 9.4 88, we have 64, 64 and 64. For 89, we have 72, 64 and 81. And for 98 we'll have 72, 81, 64. 99, we know you'll have 81, 81, and 81. And then for 9 and 10, we'll have 90, 81, and 100. OK. So we have all of the values that we need to find our sums. Because no, when we do that. Then notice that the sum of X values when we add all of these up would be equal to 49. For Y values it would be, yeah, I should actually put these in red. So we have 49 Y values it would be 54. Xy values sum would be 394. X squared value sum is 381, and the Y squared value sum is 416. Because we know that we have 9 points. OK, then we also know that N will be equal to 9. I know that we have all of this information, we can substitute it into our formula for the, the correlation coefficient. Because no, this tells us then that R is going to be equal to 9 multiplied by 394, so that's N. Multiplied by the sum of X Y minus 49 multiplied by 54. All divided OK, by the square root. Of 9 multiplied by 381. Minus 49 squared. Multiplied by the square root of 9 multiplied by 416 minus 53 square, OK? Let me just keep this for a square term outside the bracket here. No, what is this going to be? Well, if we plug this into our calculator, we should get R to be equal to 0.91799 or approximate, sorry, 91799. Let me write that properly or approximately 0.92. Since this value of correlation coefficient is close to 1, that means there is a strong positive linear correlation between the two variables. Therefore, B is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Linear Correlation Coefficient

Scatterplot

Clusters in Data

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