A random sample for five exam scores produced the following...

A random sample for five exam scores produced the following (hours of study, grade) data values: $$ \begin{array}{cc} \hline \text { Hours Studied }(x) & \text { Test Grade }(y) \\ \hline 3.5 & 88 \\ 2.4 & 76 \\ 4 & 92 \\ 5 & 85 \\ 1.1 & 60 \\ \hline \end{array} $$ a. Compute the covariance. b. Compute the correlation coefficient

A random sample for five exam scores produced the following (hours of study, grade) data values:
$$
\begin{array}{cc}
\hline \text { Hours Studied }(x) & \text { Test Grade }(y) \\
\hline 3.5 & 88 \\
2.4 & 76 \\
4 & 92 \\
5 & 85 \\
1.1 & 60 \\
\hline
\end{array}
$$
a. Compute the covariance.
b. Compute the correlation coefficient

Key Concepts

Covariance

Covariance is a statistical measure that indicates the extent to which two variables change together. If the variables tend to show similar behavior (i.e., when one increases, the other tends to increase), the covariance is positive; if they show opposite behavior, it is negative. It is computed by taking the average of the product of the deviations of each pair of data points from their respective means. This concept is fundamental in understanding relationships in bivariate data analysis.

Correlation Coefficient

The correlation coefficient is a standardized measure that quantifies the strength and direction of the linear relationship between two variables. Unlike covariance, it is dimensionless and ranges from -1 to 1, where values close to 1 or -1 imply a strong linear relationship, and values near 0 indicate little to no linear correlation. It is calculated by dividing the covariance of the two variables by the product of their standard deviations, making it easier to interpret the degree of association.

Transcript

00:01 So we have our x variable is the number of hours studied and why is the test grade? so this is ours and then this is some type of a grade, we don't know what it's out of it's 1%, don't know.

00:12 And so we have the 3.5, 45, and we certainly hope that the more you study um that you're likely to get a higher grade, but we know that there our other lurking variables in that.

00:28 So we need to find the co variance and we need to find the correlation coefficient and we're going to do this in one step, our co variance.

00:36 Remember we need to take each of our x values and subtract the mean away and each of our y values and subtract away the y, i mean and multiplied those together.

00:48 So let's find what the x bar is, the standard deviation of x.

00:52 We're going to use that for the r value, not for the co variance, but we need all this information.

00:58 So let's find it.

00:59 So i have stat calculate to variable statistics of my list one in my list too.

01:06 And let's see what we get here and we have our mean of, this is 3.2 are standard deviation is 1.5, 0.

01:16 Will call that too.

01:17 And then we have five pieces of data are mean of ry is 80.2 and our standard deviation is 12.736.

01:28 So let's put these values in.

01:30 So we're going to go back to our spreadsheet, i have this in list one, this is my list too and this is my list three.

01:38 So i have up in this component and i'll write down what these values are at least the first couple.

01:44 So i have a left parenthesis e and i'm going to put in list one minus the mean and that mean is 3.2 times and then close that parentheses off times, left parentheses.

01:58 And then my list two minus the mean of my wives, which was 80.2, whoops, 80.2...

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