Question

In: Statistics and Probability

Use the following data to answer the questions below SAT Income GPA 1651 47000 2.79 1581...

Use the following data to answer the questions below
SAT Income GPA
1651 47000 2.79
1581 34000 2.97
1790 90000 3.48
1626 60000 2.5
1754 113000 2.92
1754 71000 3.76
1706 105000 2.8
1765 59000 3.26
1786 50000 3.89
1686 27000 3.67
1790 107000 3.31
1707 109000 3.16
1804 81000 3.73
1712 62000 3.21
1607 72000 2.8
1738 63000 3.7
1790 55000 3.86
1796 64000 3.91
1547 47000 2.63
1692 89000 2.98
1711 42000 3.45
1689 70000 3.06
1740 118000 2.88
1940 113000 3.96
a What is the model for income to predict GPA
b What is the estimated GPA for someone with an income of $100,000?
c For each additional $1,000 in income, how much does their GPA increase?
d How much variation in GPA is explained by Income?

Solutions

Expert Solution

a.

        X - Mx         Y - My        (X - Mx)2 (X - Mx)(Y - My)
-25833.3333 -0.4883 667361111.1 12615.2778
-38833.3333 -0.3083 1508027778 11973.6111
17166.6667 0.2017 294694444.4 3461.9444
-12833.3333 -0.7783 164694444.4 9988.6111
40166.6667 -0.3583 1613361111 -14393.0556
-1833.3333 0.4817 3361111.111 -883.0556
32166.6667 -0.4783 1034694444 -15386.3889
-13833.3333 -0.0183 191361111.1 253.6111
-22833.3333 0.6117 521361111.1 -13966.3889
-45833.3333 0.3917 2100694444 -17951.3889
34166.6667 0.0317 1167361111 1081.9444
36166.6667 -0.1183 1308027778 -4279.7222
8166.6667 0.4517 66694444.44 3688.6111
-10833.3333 -0.0683 117361111.1 740.2778
-833.3333 -0.4783 694444.4444 398.6111
-9833.3333 0.4217 96694444.44 -4146.3889
-17833.3333 0.5817 318027777.8 -10373.0556
-8833.3333 0.6317 78027777.78 -5579.7222
-25833.3333 -0.6483 667361111.1 16748.6111
16166.6667 -0.2983 261361111.1 -4823.0556
-30833.3333 0.1717 950694444.4 -5293.0556
-2833.3333 -0.2183 8027777.778 618.6111
45166.6667 -0.3983 2040027778 -17991.3889
40166.6667 0.6817 1613361111 27380.2778
SS: 16793333333.3333 SP: -26116.6667

Sum of X = 1748000
Sum of Y = 78.68
Mean X = 72833.3333
Mean Y = 3.2783
Sum of squares (SSX) = 16793333333.3333
Sum of products (SP) = -26116.6667

Regression Equation = ŷ = bX + a

b = SP/SSX = -26116.67/16793333333.33 = 0

a = MY - bMX = 3.28 - (0*72833.33) = 3.3916

ŷ = 0X + 3.3916

b. For income of $100,000, ŷ = 3.3916

c. As here slope is 0, so value here is 0

d. To find this we will first need to find r

        X - Mx         Y - My        (X - Mx)2        (Y - My)2 (X - Mx)(Y - My)
-25833.333 -0.488 667361111.1 0.238 12615.278
-38833.333 -0.308 1508027778 0.095 11973.611
17166.667 0.202 294694444.4 0.041 3461.944
-12833.333 -0.778 164694444.4 0.606 9988.611
40166.667 -0.358 1613361111 0.128 -14393.056
-1833.333 0.482 3361111.111 0.232 -883.056
32166.667 -0.478 1034694444 0.229 -15386.389
-13833.333 -0.018 191361111.1 0 253.611
-22833.333 0.612 521361111.1 0.374 -13966.389
-45833.333 0.392 2100694444 0.153 -17951.389
34166.667 0.032 1167361111 0.001 1081.944
36166.667 -0.118 1308027778 0.014 -4279.722
8166.667 0.452 66694444.44 0.204 3688.611
-10833.333 -0.068 117361111.1 0.005 740.278
-833.333 -0.478 694444.444 0.229 398.611
-9833.333 0.422 96694444.44 0.178 -4146.389
-17833.333 0.582 318027777.8 0.338 -10373.056
-8833.333 0.632 78027777.78 0.399 -5579.722
-25833.333 -0.648 667361111.1 0.42 16748.611
16166.667 -0.298 261361111.1 0.089 -4823.056
-30833.333 0.172 950694444.4 0.029 -5293.056
-2833.333 -0.218 8027777.778 0.048 618.611
45166.667 -0.398 2040027778 0.159 -17991.389
40166.667 0.682 1613361111 0.465 27380.278
Mx: 72833.333 My: 3.278 Sum: 16793333333.333 Sum: 4.675 Sum: -26116.667

X Values
∑ = 1748000
Mean = 72833.333
∑(X - Mx)2 = SSx = 16793333333.333

Y Values
∑ = 78.68
Mean = 3.278
∑(Y - My)2 = SSy = 4.675

X and Y Combined
N = 24
∑(X - Mx)(Y - My) = -26116.667

R Calculation
r = ∑((X - My)(Y - Mx)) / √((SSx)(SSy))

r = -26116.667 / √((16793333333.333)(4.675)) = -0.0932

Now r^2=-0.0932^2=0.0087

So 8.7% of variation in GPA is explained by Income

orchestra answered 1 year ago
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