A life insurance company wishes to examine the relationship between the amount of life insurance held by a family and family income. From a random sample of 20 households, the company collected the data in the file insur.xls. The data are in thousands of dollars.
(a) Estimate a linear relationship between life insurance (Y) and income (X).
(b) Discuss the relationship you estimated in (a). In particular:
i. What is your estimate of the resulting change in the amount of life insurance when income increases by $1000?
ii. What is the standard error of the estimate in (i), and how do you use this standard error for interval estimation and hypothesis testing?
iii. One member of the management board claims that for every $1000 increase in income, the amount of life insurance held will go up by $5000. Choose an alternative hypothesis and explain your choice. Does your estimated relationship support this claim? Use a 5 percent significance level.
(c) Test the hypothesis that as income increases the amount of life insurance increase by the same amount. That is, test the hypothesis that the slope of the relationship is 1.
(d) Predict the amount of life insurance held by a family with an income of $100,000.
You can find the linear relationship (regression) in the Excel file I'm attaching. I ran the Data Analysis toolpack from Excel.
[The output is in Spanish, but I translated the relevant table -the third one- into English]
The equation that gives this regression is basically:
Insurance = 6.85 + 3.88*Income
[Insurance and Income expressed in thousands]
i) The coefficient associated to income (3.88) shows the change in insurance when income changes by $1. Therefore, for each $1 increase in income, according to this regression, insurance should increase by $3.88. Thus when income rises by $1000, insurance should rise by $3,880.
ii) The standard error of the estimate is part of the output of the regression. In this case, it is 0.1121...
We'll see how to build confidence intervals and perform hypotheses testings in the next questions. However, let's see first a very important fact that we'll use then. Let's call Beta(est) to the estimated coefficient associated to Income, Beta(pop) to the population coefficient (the one we're trying to estimate), SE to the standard error of the estimated coefficient (0.1121... in this case), and n to the sample size (in this case, 20). We have that:
Beta(est) - Beta(pop)
------------------------- follows a T distribution with n-2 degrees of freedom (df)
[obviously, 18 df in this ...
Predict the amount of life insurance.