Question

A campus researcher asked a random sample of 8 students about how many hours they work per week on average and their grade in their last math course. She then made a table for hours worked per week ? and their last math class score ?.

X	11	13	12	22	8	16	12	21
Y	97	76	80	78	87	83	81	77

∑? = 115, ∑? = 659, ∑(??) = 9344, ∑?2 = 1823, ∑?2 = 54,617

a. Compute ???, ???, and ???.

b. Compute SST, SSR, and SSE.

c. Compute the coefficient of determination, ?2. Round to four decimal places as needed.

d. Determine the percentage of variation in the observed values of the response variable explained by the regression.

e. State how useful the regression equation appears to be for making predictions. NOT VERY USEFUL or VERY USEFUL

f. Find the regression equation. Round to four decimal places as needed.

g. Predict the value of the response variable for a student working 15 hours a week.

Thank You.

Answer 1

a)

X	Y
11	97
13	76
12	80
22	78
8	87
16	83
12	81
21	77

The independent variable is X, and the dependent variable is Y.

In order to compute the regression coefficients, the following table needs to be used:

	X	Y	X*Y	X2	Y2
	11	97	1067	121	9409
	13	76	988	169	5776
	12	80	960	144	6400
	22	78	1716	484	6084
	8	87	696	64	7569
	16	83	1328	256	6889
	12	81	972	144	6561
	21	77	1617	441	5929
Sum =	115	659	9344	1823	54617

Based on the above table, the following is calculated:

Therefore, we find that the regression equation is:

b)

	X	Y	(Y - Ybar)^2
	11	97	213.890625
	13	76	40.640625
	12	80	5.640625
	22	78	19.140625
	8	87	21.390625
	16	83	0.390625
	12	81	1.890625
	21	77	28.890625
Sum =	115	659	331.875

SST =

SST = 331.875

Now that we have the regression equation, we can compute SSR​.

The regression sum of squares is computed as follows:

SSE = TSS - SSR

SSE = 331.875 - 98.1502

SSE = 233.7248

c)

Now, the correlation coefficient is computed using the following expression::

Then, the coefficient of determination, or R-Squared coefficient (R^2), is computed by simply squaring the correlation coefficient that was found above.

So we get:

Therefore, based on the sample data provided, it is found that the coefficient of determination is R^2 = 0.2957.

d)

This implies that approximately 29.57% of variation in the dependent variable is explained by the independent variable.

Therefore, based on the above calculations, the regression coefficients (the slope m, and the y-intercept n) are obtained as follows:

e)

since approximately 29.57% of variation in the dependent variable is explained by the independent variable.the equation is not that useful because of low R2.

f)

Therefore, we find that the regression equation is:

g)

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