A researcher wishes to study the relationship between education and income separately for individuals who have college degrees, and for those who don't. To this end, he interviews 100 individuals in each category. Survey results are listed in the table below. non college graduates college graduates avg education 13 yr 18 yr std. dev. education 2 yr 1.2 yr Sxx 396 yr2 143 yr2 Avg. Income $67,200 $84,950 std. dev. income $9,400 $10,500 correlation coefficient .25 .15
Inverse Normal Distribution
The method used for finding the corresponding z-critical value in a normal distribution using the known probability is said to be an inverse normal distribution. The inverse normal distribution is a continuous probability distribution with a family of two parameters.
Mean, Median, Mode
It is a descriptive summary of a data set. It can be defined by using some of the measures. The central tendencies do not provide information regarding individual data from the dataset. However, they give a summary of the data set. The central tendency or measure of central tendency is a central or typical value for a probability distribution.
Z-Scores
A z-score is a unit of measurement used in statistics to describe the position of a raw score in terms of its distance from the mean, measured with reference to standard deviation from the mean. Z-scores are useful in statistics because they allow comparison between two scores that belong to different normal distributions.
A researcher wishes to study the relationship between education and income separately for individuals who have college degrees, and for those who don't. To this end, he interviews 100 individuals in each category. Survey results are listed in the table below.
| non college graduates | college graduates | |
|---|---|---|
| avg education | 13 yr | 18 yr |
| std. dev. education | 2 yr | 1.2 yr |
| Sxx | 396 yr2 | 143 yr2 |
| Avg. Income | $67,200 | $84,950 |
| std. dev. income | $9,400 | $10,500 |
| .25 | .15 |
a. Use the data above to find point estimates for regression coefficients B0NG and B1NG for non-college graduates and BoG and B1G for college graduates.
b. Propose an unbiased estimator for the difference theta=B1G - B1NG in slope coefficients for the two sub-populations, and show that B(theta hat)= 0.
c. Assume that the error terms ENG and EG for non-graduates and graduates, respectively, are both distributed normally, with known standard deviation oENG = oEG = $
10,000. In that case, determine the standard deviation error of theta hat, along with the shape of its distribution.
d. Using the estimator defined above, and maintaining the assumption oENG = oEG = $10,000, test at the alpha = .05 level whether the slope coefficients, for the two sub-populations differ. State the p-value associated with this hypothesis test.
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