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 50 120
B 80 200
C 100 220
D 80 160
E 80 180
F 120 270
G 70 150
H 100 240
I 60 160
J 90 210
What is the least squares estimates of the slope?
What is the least squares estimate of the Y intercept?
What is the prediction for the amount of life insurance for a family whose income is $85,000?
What would be the residual (error) term for a family income of $90,000?
To find the least squares estimates of the slope and the Y intercept, we need to calculate the coefficients of the regression line using the given data.
Step 1: Calculate the means of the variables
Mean of Income (x̄) = (50 + 80 + 100 + 80 + 80 + 120 + 70 + 100 + 60 + 90) / 10 = 84
Mean of Amount of life insurance (ȳ) = (120 + 200 + 220 + 160 + 180 + 270 + 150 + 240 + 160 + 210) / 10 = 196
Step 2: Calculate the deviations from the means
Deviation of Income (x - x̄):
50 - 84 = -34
80 - 84 = -4
100 - 84 = 16
80 - 84 = -4
80 - 84 = -4
120 - 84 = 36
70 - 84 = -14
100 - 84 = 16
60 - 84 = -24
90 - 84 = 6
Deviation of Amount of life insurance (y - ȳ):
120 - 196 = -76
200 - 196 = 4
220 - 196 = 24
160 - 196 = -36
180 - 196 = -16
270 - 196 = 74
150 - 196 = -46
240 - 196 = 44
160 - 196 = -36
210 - 196 = 14
Step 3: Calculate the product of the deviations
Product of Deviations (x - x̄) * (y - ȳ):
-34 * -76 = 2,584
-4 * 4 = 16
16 * 24 = 384
-4 * -36 = 144
-4 * -16 = 64
36 * 74 = 2,664
-14 * -46 = 644
16 * 44 = 704
-24 * -36 = 864
6 * 14 = 84
Step 4: Calculate the squared deviations of Income
Squared Deviation of Income (x - x̄)^2:
(-34)^2 = 1,156
(-4)^2 = 16
(16)^2 = 256
(-4)^2 = 16
(-4)^2 = 16
(36)^2 = 1,296
(-14)^2 = 196
(16)^2 = 256
(-24)^2 = 576
(6)^2 = 36
Step 5: Calculate the least squares estimates of the slope (β1) and the Y intercept (β0)
β1 = Sum of (x - x̄) * (y - ȳ) / Sum of (x - x̄)^2
β1 = (2,584 + 16 + 384 + 144 + 64 + 2,664 + 644 + 704 + 864 + 84) / (1,156 + 16 + 256 + 16 + 16 + 1,296 + 196 + 256 + 576 + 36)
β1 = 7,960 / 4,372
β1 ≈ 1.818
β0 = ȳ - (β1 * x̄)
β0 = 196 - (1.818 * 84)
β0 ≈ 26.632
Therefore, the least squares estimate of the slope (β1) is approximately 1.818 and the least squares estimate of the Y intercept (β0) is approximately 26.632.
To find the predicted amount of life insurance for a family with an income of $85,000, we can use the regression line equation:
y = β0 + β1 * x
where:
y = predicted amount of life insurance
x = income
Plugging in the values:
y = 26.632 + 1.818 * 85
y ≈ 26.632 + 154.63
y ≈ 181.262
Therefore, the predicted amount of life insurance for a family with an income of $85,000 is approximately $181,262.
To find the residual (error) term for a family income of $90,000, we can use the regression line equation:
y = β0 + β1 * x
where:
y = predicted amount of life insurance
x = income
Plugging in the values:
y = 26.632 + 1.818 * 90
y ≈ 26.632 + 163.62
y ≈ 190.252
The residual (error) term is the difference between the actual amount of life insurance and the predicted amount of life insurance:
Residual = Actual Amount of life insurance - Predicted Amount of life insurance
Residual = Actual Amount of life insurance - 190.252
As we do not have the actual amount of life insurance for a family with an income of $90,000, we cannot calculate the residual term.
To find the least squares estimates of the slope and the Y-intercept, we need to perform linear regression analysis on the given data.
Step 1: Calculate the means of the income (X) and the amount of life insurance (Y).
Mean of X = (50 + 80 + 100 + 80 + 80 + 120 + 70 + 100 + 60 + 90) / 10 = 84
Mean of Y = (120 + 200 + 220 + 160 + 180 + 270 + 150 + 240 + 160 + 210) / 10 = 195
Step 2: Calculate the deviations from the mean for both X and Y.
X deviations: 50 - 84, 80 - 84, 100 - 84, 80 - 84, 80 - 84, 120 - 84, 70 - 84, 100 - 84, 60 - 84, 90 - 84
Y deviations: 120 - 195, 200 - 195, 220 - 195, 160 - 195, 180 - 195, 270 - 195, 150 - 195, 240 - 195, 160 - 195, 210 - 195
Step 3: Calculate the product of X deviations and Y deviations.
Product of deviations:
(50 - 84)(120 - 195), (80 - 84)(200 - 195), (100 - 84)(220 - 195), (80 - 84)(160 - 195), (80 - 84)(180 - 195),
(120 - 84)(270 - 195), (70 - 84)(150 - 195), (100 - 84)(240 - 195), (60 - 84)(160 - 195), (90 - 84)(210 - 195)
Step 4: Calculate the sum of the product of deviations.
Sum of products: Sum of all calculated product of deviations
Step 5: Calculate the sum of squared X deviations.
Sum of squared X deviations: (50 - 84)^2 + (80 - 84)^2 + (100 - 84)^2 + (80 - 84)^2 + (80 - 84)^2 + (120 - 84)^2 + (70 - 84)^2 + (100 - 84)^2 + (60 - 84)^2 + (90 - 84)^2
Step 6: Calculate the least squares estimate of the slope.
Slope (b) = Sum of products / Sum of squared X deviations
Step 7: Calculate the least squares estimate of the Y-intercept.
Y-intercept (a) = Mean of Y - (Slope * Mean of X)
Step 8: Calculate the prediction for the amount of life insurance for a family whose income is $85,000.
Prediction = Y-intercept + (Slope * $85,000)
Step 9: Calculate the residual (error) term for a family income of $90,000.
Residual term = Actual value - Predicted value
Now, follow these steps and apply the formulas to the given data to find the answers to each question.