Question

You are given the following information about x and y.

x y Independent Dependent Variable Variable 15 5 12 7 10 9 7 11
The least squares estimate of b 0 equals ______.
a. 16.41176
b. â1.3
c. 21.4
d. â7.647

Answer 1

`b_0 = 16.471`

Explanation:

Given

`\begin{array}{ccccc}x & {15} & {12} & {10} & {7} \ \\ y & {5} & {7} & {9} & {11} \ \end{array}`

Required

The least square estimate
`b_0`

Calculate the mean of x

`\bar x = (\sum x)/(n)`

`\bar x = (15+12+10+7)/(4) =(44)/(4) = 11`

Calculate the mean of y

`\bar y = (\sum y)/(n)`

`\bar y = (5+7+9+11)/(4) =(32)/(4) = 8`

Calculate
`\sum(x - \bar x) * (y - \bar y)`

`\sum(x - \bar x) = (15 - 11) * (5 - 8)+ (12 - 11) * (7 - 8) + (10 - 11) * (9 - 8)+ (7 - 11) * (11 - 8)`

`\sum(x - \bar x) = -26`

Calculate
`\sum(x - \bar x)^2`

`\sum(x - \bar x)^2 = (15 - 11)^2 + (12 - 11)^2 + (10 - 11)^2 + (7 - 11)^2`

`\sum(x - \bar x)^2 = 34`

So:

`b = (\sum(x - \bar x) * (y - \bar y))/(\sum(x - \bar x)^2)`

`b = (-26)/(34)`

`b_0 = y - bx`

`b_0 = 5 - (-26)/(34)*15`

`b_0 = 5 + (26*15)/(34)`

`b_0 = 5 + (390)/(34)`

Take LCM

`b_0 = (34*5+ 390)/(34)`

`b_0 = (560)/(34)`

`b_0 = 16.471`