Question

1. a linear regression was performed using the five following data points: a(1, 11), b(2, 7), c(5, 2), d(7, 1), e(9, - 2). the residual for which of the five points has the largest impact on the lsrl? (use your calculator!!)

(a) point a
(b) point b
(c) point c
(d) point d
(e) point e

Explanation:

Step1: Input data into calculator

Input the \(x\) - values (\(1,3,5,7,9\)) and \(y\) - values (\(11,7,3,1, - 2\)) into the linear regression function of the calculator.

Step2: Calculate predicted \(y\) - values

For a linear regression model \(y = a+bx\), after running the regression, for \(x = 1\), \(y_{predicted}=a + b\times1\); for \(x = 3\), \(y_{predicted}=a + b\times3\); for \(x = 5\), \(y_{predicted}=a + b\times5\); for \(x = 7\), \(y_{predicted}=a + b\times7\); for \(x = 9\), \(y_{predicted}=a + b\times9\).

Step3: Calculate residuals

Residual \(e=y - y_{predicted}\).
For point \(A(1,11)\): \(e_1=11-(a + b\times1)\)
For point \(B(3,7)\): \(e_2=7-(a + b\times3)\)
For point \(C(5,3)\): \(e_3=3-(a + b\times5)\)
For point \(D(7,1)\): \(e_4=1-(a + b\times7)\)
For point \(E(9,-2)\): \(e_5=-2-(a + b\times9)\)
The point with the largest residual (in absolute value) has the largest impact on the least - squares regression line (LSRL).
Using a TI - 84 (or similar calculator):
Enter \(L_1=\{1,3,5,7,9\}\) and \(L_2 = \{11,7,3,1,-2\}\).
Run LinReg \(y=ax + b\). We get \(y=-1.5x+12.5\)
For \(x = 1\): \(y_{predicted}=-1.5\times1 + 12.5=11\), \(e_1=11 - 11=0\)
For \(x = 3\): \(y_{predicted}=-1.5\times3+12.5 = 8\), \(e_2=7 - 8=-1\)
For \(x = 5\): \(y_{predicted}=-1.5\times5+12.5=5\), \(e_3=3 - 5=-2\)
For \(x = 7\): \(y_{predicted}=-1.5\times7+12.5=2\), \(e_4=1 - 2=-1\)
For \(x = 9\): \(y_{predicted}=-1.5\times9+12.5=-1\), \(e_5=-2-(-1)=-1\)

Answer:

C. Point C