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.
163
To answer these questions, we need to perform a linear regression analysis on the given data. This will help us estimate the relationship between income and the amount of life insurance held by families.
Let's start by calculating the least squares estimate of the slope.
Question 1:
The formula for the slope (b) is given by:
b = Σ((xi - x̄)(yi - ȳ)) / Σ((xi - x̄)²)
where xi represents the income, yi represents the amount of life insurance, x̄ is the mean of the income, and ȳ is the mean of the amount of life insurance.
First, calculate the mean of the income (x̄) and the mean of the amount of life insurance (ȳ).
x̄ = (40 + 80 + 110 + 80 + 80 + 120 + 60 + 100 + 60 + 90) / 10 = 80
ȳ = (110 + 200 + 220 + 150 + 170 + 270 + 140 + 240 + 150 + 200) / 10 = 182
Next, calculate the numerator and denominator of the slope formula.
Numerator: Σ((xi - x̄)(yi - ȳ))
= (40 - 80)(110 - 182) + (80 - 80)(200 - 182) + ... + (90 - 80)(200 - 182)
= -160 * -72 + 0 * 18 + ... + 10 * 18
= 11520 + 0 + ... + 180
= 14020
Denominator: Σ((xi - x̄)²)
= (40 - 80)² + (80 - 80)² + ... + (90 - 80)²
= 1600 + 0 + ... + 100
= 3900
Now, calculate the least squares estimate of the slope (b).
b = Σ((xi - x̄)(yi - ȳ)) / Σ((xi - x̄)²)
= 14020 / 3900
≈ 3.5897
Therefore, the least squares estimate of the slope is approximately 3.5897.
Question 2:
The formula for the Y-intercept (a) is given by:
a = ȳ - b * x̄
where ȳ is the mean of the amount of life insurance, b is the slope, and x̄ is the mean of the income.
Using the values we calculated earlier:
ȳ = 182
b ≈ 3.5897
x̄ = 80
a = 182 - 3.5897 * 80
≈ 182 - 287.176
≈ -105.176
Therefore, the least squares estimate of the Y-intercept is approximately -105.176.
Question 3:
To predict the amount of life insurance for a family whose income is $85,000, we can use the equation of the regression line:
ŷ = a + b * x
where ŷ is the predicted amount of life insurance, a is the Y-intercept, b is the slope, and x is the given income.
Using the values we calculated earlier:
a ≈ -105.176
b ≈ 3.5897
x = 85 (thousands of dollars)
ŷ = -105.176 + 3.5897 * 85
≈ -105.176 + 305.9245
≈ 200.7485
Therefore, the prediction for the amount of life insurance for a family whose income is $85,000 is approximately $200,748.50.
Question 4:
To find the residual (error) term for a family income of $90,000, we can use the regression equation:
ŷ = a + b * x
First, calculate the predicted amount of life insurance for an income of $90,000 using the values we calculated earlier:
a ≈ -105.176
b ≈ 3.5897
x = 90 (thousands of dollars)
ŷ = -105.176 + 3.5897 * 90
≈ -105.176 + 323.073
≈ 217.897
Now, calculate the residual term (error):
Residual = Actual value - Predicted value
In this case, the actual amount of life insurance is not given. Therefore, we cannot calculate the specific residual term for a family income of $90,000 without additional information.
To calculate residuals for other data points, subtract the predicted value (ŷ) from the actual value (yi) for each data point.
I hope this helps!