Following is a random sample of five (x, y) pairs of data points: (12,200)(30,600)(15,270)(24,50

step 1 So on this problem, we have a set of five pieces of data. And we have our values of 12, 30, 15,

Following is a random sample of five (x, y) pairs of data...

Key Concepts

Covariance

Covariance is a statistical measure that quantifies the degree to which two variables change together. It is calculated as the average product of the deviations of each data point from their respective means. The sign of the covariance indicates the type of relationship; a positive covariance means that as one variable increases, the other tends to increase as well, while a negative covariance indicates an inverse relationship.

Correlation Coefficient

The correlation coefficient, often referred to as Pearson's r, is a standardized measure of the linear relationship between two variables. It is computed by dividing the covariance of the two variables by the product of their standard deviations. The resulting value falls between -1 and 1, where values close to 1 or -1 signify a strong linear relationship, and values near 0 indicate a weak linear association.

Transcript

00:01 So on this problem, we have a set of five pieces of data.

00:04 And we have our values of 12, 30, 15, 24, and 14 as our x values, which i'm putting those in list one.

00:15 And our y values that i'm putting in list two, we have the 200, 600, 270, 500, and 210.

00:28 And we want to look for, in part a, we want to find our covariance, which our covariance will end up being, we have to find what the sum of the values of the x minus the x bar times the y minus the y bar, and then we will divide that by n minus one.

00:51 And then we'll find the correlation coefficient.

00:53 So let's do that, and we'll use our spreadsheet capacity in order to find those values.

00:57 And the first thing we need to do is after we enter those values, we do need to find what our mean of our x is, the mean of our y, our standard deviation of our x, not for the covariance, but for the correlation coefficient, and the standard deviation of the y.

01:12 And if i use two variable statistics on my calculator, i can get what those are just like that.

01:20 And so this is convenient.

01:21 That comes out to be 19.

01:23 The standard deviation is 7 .681.

01:27 And for our y bar, we have our y bar is 356 with a standard deviation of 182 .5655.

01:41 And notice that will be a lot larger because there's a lot more variability on those y values.

01:45 So next we need to figure out what are x of each of these, x minus the x bar and the y minus the y bar.

01:57 All of these values down.

01:59 I'll just put the first cell.

02:01 And i'm going to put that into my list three.

02:03 And when i type it into my list three, i'm going to type in list one minus the x bar.

02:08 And i can find that x bar as well, i can just type in 19.

02:12 I guess that would be just as easy.

02:14 So we have stat.

02:16 And we're going to take our list one minus our x bar, which we just said was 19.

02:23 And so our first value is negative 7 and the last phase is negative 5.

02:28 So again, i'm not going to write them all down.

02:30 And then for this one, we're going to take our list 2 minus our y bar, which we can use this decimal.

02:36 Well, it's not a decimal approximation, so we can just use the 356...

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