An insurance company wishes to examine the relationship between income (in $,000) and the amount of life insurance (in $,000) held by families. The company drew a simple random sample of families and obtained the following results:
Family - Income - Amount of Life Insurance
A - 40 - 110
B - 80 - 200
C - 110 - 220
D - 80 - 150
E - 80 - 170
F - 120 - 270
G - 60 - 140
H - 100 - 240
I - 60 - 150
J - 90 - 200
Question 1
What is the least squares estimate of the slope?
Question 2
What is the least squares estimate of the Y intercept?
Question 3
What is the prediction for the amount of life insurance for a family whose income is $85,000?
Question 4
What would be the residual (error) term for a family income of $90,000?
All answers should be to four decimal places.
1.9216
27.4254
163.3634
-0.3731
To find the least squares estimate of the slope and the Y-intercept, we need to perform linear regression analysis on the given data.
Question 1: What is the least squares estimate of the slope?
To find the slope, we need to calculate the sum of products of the deviations. Let's go step by step:
Step 1: Calculate the mean of the income and the amount of life insurance:
Mean of income (X̄) = (40 + 80 + 110 + 80 + 80 + 120 + 60 + 100 + 60 + 90) / 10 = 78
Mean of life insurance (Ȳ) = (110 + 200 + 220 + 150 + 170 + 270 + 140 + 240 + 150 + 200) / 10 = 187
Step 2: Calculate the deviations for each data point:
Deviation for income (Xi - X̄):
A: 40 - 78 = -38
B: 80 - 78 = 2
C: 110 - 78 = 32
D: 80 - 78 = 2
E: 80 - 78 = 2
F: 120 - 78 = 42
G: 60 - 78 = -18
H: 100 - 78 = 22
I: 60 - 78 = -18
J: 90 - 78 = 12
Deviation for life insurance (Yi - Ȳ):
A: 110 - 187 = -77
B: 200 - 187 = 13
C: 220 - 187 = 33
D: 150 - 187 = -37
E: 170 - 187 = -17
F: 270 - 187 = 83
G: 140 - 187 = -47
H: 240 - 187 = 53
I: 150 - 187 = -37
J: 200 - 187 = 13
Step 3: Calculate the sum of products of the deviations:
∑((Xi - X̄) * (Yi - Ȳ)) = (-38 * -77) + (2 * 13) + (32 * 33) + (2 * -37) + (2 * -17) + (42 * 83) + (-18 * -47) + (22 * 53) + (-18 * -37) + (12 * 13) = 23682
Step 4: Calculate the sum of squared deviations for the income variable:
∑((Xi - X̄)^2) = (-38)^2 + 2^2 + 32^2 + 2^2 + 2^2 + 42^2 + (-18)^2 + 22^2 + (-18)^2 + 12^2 = 7132
Step 5: Calculate the least squares estimate of the slope (b):
b = ∑((Xi - X̄) * (Yi - Ȳ)) / ∑((Xi - X̄)^2) = 23682 / 7132 ≈ 3.3201
Therefore, the least squares estimate of the slope is approximately 3.3201.
Question 2: What is the least squares estimate of the Y-intercept?
To find the Y-intercept (a), we can use the formula:
a = Ȳ - b * X̄
Using the mean of income (X̄) and life insurance (Ȳ) calculated previously, substituting the values:
a = 187 - 3.3201 * 78 ≈ 9.9936
Therefore, the least squares estimate of the Y-intercept is approximately 9.9936.
Question 3: What is the prediction for the amount of life insurance for a family whose income is $85,000?
To predict the amount of life insurance for a specific income value, we can use the regression equation:
Y = a + b * X
Substituting the values into the equation, where X is the income of $85,000:
Y = 9.9936 + 3.3201 * 85 ≈ 285.3741
Therefore, the prediction for the amount of life insurance for a family with an income of $85,000 is approximately $285,374.10.
Question 4: What would be the residual (error) term for a family income of $90,000?
To find the residual term, we use the regression equation and subtract the predicted value from the actual value.
Using the regression equation:
Y = a + b * X
Substituting the values, where X is the income of $90,000 and Y is the actual value (from the given data):
Y_actual = 240
Calculating the predicted value using the regression equation:
Y_predicted = 9.9936 + 3.3201 * 90 ≈ 301.2176
Residual (error) term = Y_actual - Y_predicted = 240 - 301.2176 ≈ -61.2176
Therefore, the residual term for a family income of $90,000 is approximately -61.2176.