The data from data491.dat contains information on 78 seventh-grade students. We

Question

The data from data491.dat contains information on 78 seventh-grade students. We want to know how well each of IQ score and self-concept score predicts GPA using least-squares regression. We also want to know which of these explanatory variables predicts GPA better. Give numerical measures that answer these questions. (Round your answers to three decimal places.)

obs gpa iq gender concept 1 7.94 113 2 56 2 8.292 107 2 75 3 4.643 96 2 59 4 7.47 116 2 79 5 8.882 107 1 78 6 7.585 117 2 49 7 7.65 100 2 55 8 2.412 79 2 44 9 6 120 1 54 10 8.833 121 2 63 11 7.47 125 1 55 12 5.528 95 1 45 13 7.167 108 2 60 14 7.571 108 1 37 15 4.7 110 1 54 16 8.167 110 1 44 17 7.822 109 1 54 18 7.598 114 1 66 19 4 85 2 50 20 6.231 107 1 54 21 7.643 112 2 49 22 1.76 80 2 5 24 6.419 102 1 65 26 9.648 137 2 64 27 10.7 118 1 65 28 10.58 123 2 68 29 9.429 132 2 51 30 8 120 2 64 31 9.585 144 2 40 32 9.571 98 1 77 33 8.998 109 1 53 34 8.333 119 1 49 35 8.175 110 2 77 36 8 114 2 45 37 9.333 120 1 61 38 9.5 118 2 71 39 9.167 89 2 64 40 10.14 127 1 57 41 9.999 113 1 76 43 10.76 110 2 75 44 9.763 118 2 65 45 9.41 119 2 52 46 9.167 117 2 65 47 9.348 105 2 95 48 8.167 93 2 60 50 3.647 98 2 41 51 3.408 69 1 43 52 3.936 87 2 60 53 7.167 98 2 55 54 7.647 117 2 52 55 0.53 59 2 38 56 6.173 98 2 52 57 7.295 117 2 56 58 7.295 114 1 43 59 8.938 117 1 76 60 7.882 91 1 44 61 8.353 120 2 57 62 5.062 92 2 53 63 8.175 113 2 61 64 8.235 103 2 72 65 7.588 111 2 28 68 7.647 97 2 59 69 5.237 97 1 48 71 7.825 124 2 54 72 7.333 91 1 53 74 9.167 110 2 60 76 7.996 98 2 59 77 8.714 97 1 44 78 7.833 102 1 51 79 4.885 89 2 47 80 7.998 91 1 55 83 3.82 101 2 38 84 5.936 114 1 69 85 9 112 1 49 86 9.5 101 1 67 87 6.057 104 2 62 88 6.057 109 1 47

Explanation / Answer

Here, response variable is GPA & explanatory variables are iq & concept

R-code for fit least-squares regression between gpa & iq is:

data=read.csv("G:/data.csv",header=TRUE)
names(data)
attach(data)
m1=lm(gpa~iq)
summary(m1)

Output:

Call:
lm(formula = gpa ~ iq)

Residuals:
Min 1Q Median 3Q Max
-3.0586 -0.8952 -0.1589 1.1770 3.5101

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -3.25014 1.32951 -2.445 0.0168 *
iq 0.10008 0.01232 8.123 6.97e-12 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.551 on 75 degrees of freedom
Multiple R-squared: 0.468, Adjusted R-squared: 0.4609
F-statistic: 65.98 on 1 and 75 DF, p-value: 6.971e-12

m2=lm(gpa~concept)

R-code for fit  least-squares regression between gpa & concept is:

data=read.csv("G:/data.csv",header=TRUE)
names(data)
attach(data)
m2=lm(gpa~concept)
summary(m2)

Output:

Call:
lm(formula = gpa ~ concept)

Residuals:
Min 1Q Median 3Q Max
-5.3311 -1.0744 0.3157 1.3945 3.5502

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.55958 0.89360 2.864 0.00542 **
concept 0.08688 0.01545 5.624 3.05e-07 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.784 on 75 degrees of freedom
Multiple R-squared: 0.2966, Adjusted R-squared: 0.2872
F-statistic: 31.62 on 1 and 75 DF, p-value: 3.051e-07

Note that here p-value corresponding to both variables is less than 0.05 ,hence these two variables iq & concept are statistically significant .

Adjusted R-squared= 0.4609 for iq &

Adjusted R-squared= 0.2872 for concept.

Since Adjusted R-squared= 0.4609 for iq > Adjusted R-squared= 0.2872 for concept , variable iq explains better variation in GPA than variable concept.

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