For this question, please use the formula sheet and like terms (no x bar etc). A

Question

For this question, please use the formula sheet and like terms (no x bar etc). A professor obtains SAT scores and freshman grade point averages (GPAs) for a group of n = 15 college students. The SAT scores have a mean of M = 580 with SS = 22,400, and the GPAs have a mean of 3.1 with SS = 1.26, and SP = 84.

Find the regression equation for predicting GPA from SAT scores.

What percentage of the variance in GPAs is accounted for by the regression equation? (Compute the correlation, r, then find r2.)

Does the regression equation account for a significant portion of the variance in GPA? Use = .05 to evaluate the F-ratio.

Explanation / Answer

here SSx = 22400, SSy = 1.26 and SSxy(SP) = 84, X-bar = 580 and Y-bar = 3.1,

DEFINITIONS:
b1 - This is the SLOPE of the regression line. Thus this is the amount that the Y variable (dependent) will change for each 1 unit change in the X variable.
b0 - This is the intercept of the regression line with the y-axis. In otherwords it is the value of Y if the value of X = 0.
Y-hat = b0 + b1(x) - This is the sample regression line. You must calculate b0 & b1 to create this line. Y-hat stands for the predicted value of Y, and it can be obtained by plugging an individual value of x into the equation and calculating y-hat.

We will solve it step by step;

Step 1):

find b1 - One method of caluating b1 is b1 = SSxy/SSx =84/22400= 2765/85 =0.00375. This is the slope of the line - for every unit change in X, y will increase by 0.00375. It is a positive number, thus its a direct relationship - as X goes up, so does Y. However, if b1 = -0.00375, then we would know the relationship between X & Y is an inverse relationship - as X goes up, y goes down)
Step2):

Find b0 - again the formula is b0 = y-bar - b1(x-bar) =1.26-0.00375*580 = 1.26-2.175 = -0.915, this is the intercept of the line and the Y-axis, and can be interpreted as the value of Y if zero hours of overtime (x=0) are worked.
(3) Create Line - Y-hat = b0 + b1(x) or Y-hat = -0.915 + 0.00375(x), This line quantifies the relationship between X & Y.

Correlation r = b1*(SSx/SSy)

=0.00375*(22400/1.26) = 66.66. Correlation value must be between -1 to 1.Plesae check your all the values given in question and submit the question.

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