Question

6. I've been doing research on the leaders of our country, and found the educational records of several US senators. I want to know if there is a connection between ACT scores and College GPA. Use the following data to answer the questions. ACT Score College GPA 16 1.85 Senator Shadow Moon Sheev Palpatine Pippin Took Locke Lamora 18 2.2 24 2.8 25 3.5 34 4.0 27 3.18 29 3.9 Wilhuff Tarkin Herbert Camacho Harry Dresden Dalinar Kholin Tom Bombadil James Howlett 25 2.9 30 4.0 21 2.6 a. Draw a scatter plot of the data, letting ACT score be your explanatory variable. Be sure to label your axis! b. Find the correlation coefficient between ACT score and college GPA. c. Does a linear correlation exist? Explain! d. Find the equation of the least-squares regression line. Show your work! e. Predict the ACT score of someone who got a 3.0 in college. f. Compute the residual for a senator with an ACT score of 27. g. Interpret the slope, if appropriate. If not, explain why. h. Interpret the y-intercept, if appropriate. If not, explain why. i. What is the coefficient of determination? j. Explain your coefficient of determination in context.

Answer 1

(a) The scatterplot is:

(b) r = 0.952

(c) Yes, there is a positive linear relationship between the variables.

(d) The regression line is:

y = 0.131 x - 0.174

The calculations are:

ACT Score	College GPA	(x - xbar)	(y - ybar)	(x - xbar)*(y - ybar)	(x - xbar)²	(y - ybar)²
16	1.85	-8.9	-1.243	11.0627	79.21	1.545049
18	2.2	-6.9	-0.893	6.1617	47.61	0.797449
24	2.8	-0.9	-0.293	0.2637	0.81	0.085849
25	3.5	0.1	0.407	0.0407	0.01	0.165649
34	4	9.1	0.907	8.2537	82.81	0.822649
27	3.18	2.1	0.087	0.1827	4.41	0.007569
29	3.9	4.1	0.807	3.3087	16.81	0.651249
25	2.9	0.1	-0.193	-0.0193	0.01	0.037249
30	4	5.1	0.907	4.6257	26.01	0.822649
21	2.6	-3.9	-0.493	1.9227	15.21	0.243049
xbar	ybar	Σ(x - xbar)	Σ(y - ybar)	Σ(x - xbar)*(y - ybar)	Σ(x - xbar)²	Σ(y - ybar)²
24.9	3.093	0	0	35.803	272.9	5.17841
r = Σ(x - xbar)*(y - ybar)/√Σ(x - xbar)²*Σ(y - ybar)²	0.9524
b1 = Σ(x - xbar)*(y - ybar)/Σ(x - xbar)²	0.131195
bo = ybar - b1*xbar	-0.17374