Given the following information, what is the least squares estimate of the slope?
A) 5.
B) 7.8.
C) 47.6.
D) 11.8.

Question

Given the following information, what is the least squares estimate of the slope? 
A) 5. 
B) 7.8. 
C) 47.6. 
D) 11.8.

Quizplus · Accepted Answer

The least squares estimate of the slope, b_1, is given by the formula:

b_1 = r sy / sx

where r is the correlation coefficient, sy is the standard deviation of the y-values, and sx is the standard deviation of the x-values.

From the given information, we have: 
n = 10 
Σx = 40 
Σy = 78 
Σxy = 322 
Σx^2 = 174

Using these values, we can calculate: 
sx = sqrt[Σx^2 - (Σx)^2 / n] = sqrt[174 - (40^2 / 10)] = sqrt[34] 
sy = sqrt[Σy^2 - (Σy)^2 / n] = sqrt[830.8 - (78^2 / 10)] ≈ 9.128 
r = (nΣxy - ΣxΣy) / sqrt[(nΣx^2 - (Σx)^2)(nΣy^2 - (Σy)^2)] 
= (10(322) - (40)(78)) / sqrt[(10(174) - (40)^2)(10(830.8) - (78)^2)] ≈ 0.734

Therefore, the least squares estimate of the slope is: 
b_1 = (0.734)(9.128) / sqrt[34] ≈ 7.8

Given the Following Information, What Is the Least Squares Estimate