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, 80, 120
B, 100, 200
C, 110, 220
D, 90, 160
E, 80, 180
F, 140, 270
G, 110, 150
H, 100, 240
I, 80, 160
J, 100, 210
QUESTION a
What is the least squares estimate of the slope?
Answer should be to four decimal places e.g. 1.2345.
QUESTION b
What is the least squares estimate of the Y intercept?
Answer should be to four decimal places e.g. 1.2345.
QUESTION c
What is the prediction for the amount of life insurance for a family whose income is $85,000?
QUESTION d
What would be the residual (error) term for a family income of $90,000?
To find the least squares estimate of the slope and Y-intercept, as well as the prediction for a given income and the residual term, we need to perform linear regression analysis. Here's how you can find the answers to each question:
QUESTION a: Estimate of the slope
1. Start by calculating the means of the income (X) and the amount of life insurance (Y).
- Mean of X (income) = (80 + 100 + 110 + 90 + 80 + 140 + 110 + 100 + 80 + 100) / 10 = 99
- Mean of Y (amount of life insurance) = (120 + 200 + 220 + 160 + 180 + 270 + 150 + 240 + 160 + 210) / 10 = 187
2. Calculate the differences between each X value and the mean of X (income - X̄) and the differences between each Y value and the mean of Y (amount of life insurance - Ȳ).
- Difference between X and X̄: (80 - 99), (100 - 99), (110 - 99), (90 - 99), (80 - 99), (140 - 99), (110 - 99), (100 - 99), (80 - 99), (100 - 99)
- Difference between Y and Ȳ: (120 - 187), (200 - 187), (220 - 187), (160 - 187), (180 - 187), (270 - 187), (150 - 187), (240 - 187), (160 - 187), (210 - 187)
3. Calculate the sum of the products of the differences above.
- Sum of (income - X̄) * (amount of life insurance - Ȳ) = (80 - 99) * (120 - 187) + (100 - 99) * (200 - 187) + ... + (100 - 99) * (210 - 187)
4. Calculate the sum of the squared differences of X (income).
- Sum of (income - X̄)^2 = (80 - 99)^2 + (100 - 99)^2 + ... + (100 - 99)^2
5. Calculate the least squares estimate of the slope (b).
- b = sum of (income - X̄) * (amount of life insurance - Ȳ) / sum of (income - X̄)^2
6. Round the least squares estimate of the slope to four decimal places.
QUESTION b: Estimate of the Y-intercept
1. Using the means calculated in QUESTION a, apply the least squares estimate formula.
- b = sum of (income - X̄) * (amount of life insurance - Ȳ) / sum of (income - X̄)^2
2. Calculate the Y-intercept (a) using the formula:
- a = Ȳ - b * X̄
3. Round the least squares estimate of the Y-intercept to four decimal places.
QUESTION c: Prediction for an income of $85,000
1. Use the Y = a + bX formula with the values obtained from QUESTION b.
- Income (X) = 85 (thousand dollars)
2. Substitute the X value into the formula:
- Y = a + bX
3. Round the predicted amount of life insurance (Y) to the nearest dollar.
QUESTION d: Residual (error) term for an income of $90,000
1. Use the predicted equation obtained in QUESTION c.
- Y = a + bX
2. Substitute the X value of $90,000:
- X = 90
3. Calculate the predicted amount of life insurance (Y) for an income of $90,000.
4. Calculate the residual term by subtracting the predicted amount of life insurance (Y) from the actual amount of life insurance for the given income.
Now, follow these steps and calculations to find the answers to each question.