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?
Well, this is regression analysis... Have a look at the wikipedia article.
To find the least squares estimates of the slope and Y intercept, we need to perform linear regression using the given data. Here's how you can do it:
Step 1: Calculate the means of the income (x) and amount of life insurance (y) values:
- Mean of income (x): (50 + 80 + 100 + 80 + 80 + 120 + 70 + 100 + 60 + 90) / 10 = 84
- Mean of amount of life insurance (y): (120 + 200 + 220 + 160 + 180 + 270 + 150 + 240 + 160 + 210) / 10 = 198
Step 2: Calculate the deviations from the means for both x and y values. Deviation = value - mean.
- Deviations for x: (50 - 84), (80 - 84), (100 - 84), (80 - 84), (80 - 84), (120 - 84), (70 - 84), (100 - 84), (60 - 84), (90 - 84)
- Deviations for y: (120 - 198), (200 - 198), (220 - 198), (160 - 198), (180 - 198), (270 - 198), (150 - 198), (240 - 198), (160 - 198), (210 - 198)
Step 3: Calculate the sum of the products of the deviations of x and y. Sum of (x - mean of x) * (y - mean of y).
- Sum of products of deviations: (50 - 84) * (120 - 198) + (80 - 84) * (200 - 198) + ... + (90 - 84) * (210 - 198)
Step 4: Calculate the sum of the squared deviations of x. Sum of (x - mean of x)^2.
- Sum of squared deviations of x: (50 - 84)^2 + (80 - 84)^2 + ... + (90 - 84)^2
Step 5: Calculate the least squares estimate of the slope (b) using the formula:
b = sum of products of deviations / sum of squared deviations of x
Step 6: Calculate the least squares estimate of the Y intercept (a) using the formula:
a = mean of y - b * mean of x
Once you have determined the slope (b) and the Y intercept (a), you can answer the remaining questions:
- The least squares estimate of the slope is the value of b from Step 5.
- The least squares estimate of the Y intercept is the value of a from Step 6.
To predict the amount of life insurance for a family whose income is $85,000, you need to substitute the income value (85) into the equation of the regression line: y = a + b*x, where x is the income and y is the predicted amount of life insurance.
To calculate the residual (error) term for a family income of $90,000, you would use the predicted amount of life insurance using the regression line (as mentioned in the previous step) and subtract the actual amount of life insurance obtained from the data. The residual term represents the difference between the predicted and actual values for the given input (family income of $90,000).