A sociologist is interested in the relationship between the median income of a j

Question

A sociologist is interested in the relationship between the median income of a jurisdiction and its high school dropout rate. S/he estimates a regression equation with median income as the dependent variable and percent of the adult population who are high school dropouts as the independent variable for a random sample of jurisdictions, yielding the following results: Y = 62, 600 - 857 (X) where Y = predicted median income (in dollars); X = percent of adult population who have dropped out of high school; the standard error (se) of the slope = 207 (a) What is the predicted change in median income as the high school dropout measure increases by one unit. i.e. what is the linear "effect" of the high school dropout measure? (b) What is the predicted median income for a jurisdiction with a value of 10 on the dropout measure? Show your work. (c) Is the linear relationship between median income and the level of dropouts statically significant at the .05 level? Explain the basis of your answer. Formula: Confidence Interval (at 95% confidence level) =b plusminus 2*sc

Explanation / Answer

ANSWER

Back-up Theory

A general linear regression is given by: y = a + bx, where y is the dependent variable and x is the independent variable. b, called the regression coefficient of y on x, represents the expected change in y per unit change in x. Since b is sign sensitive, a positive b implies x and y move in the same direction while a negaitive b implies x and y move in the opposite direction.

Now, to answer the question,

in the given problem, y is the median income and x is the drop out measure.

Given y = 62600 - 857x, if drop-out measure increases by 1 unit, the predicted value of median incime would come down by $857.

Drop-out measure is10 implies x =10 and hence the corresponding predicted value of median income would be:

y = 62600 - (857 x 10) = 62600 - 8570 = 54030

To test if the obtained linear regression represents a statistically significant relationship between x and y, we test whether actual value of b could be zero. This can be done either by testing theenull hypothesis H0 : b = 0 vs the alternative H1: b is not equal to zero or checking whether the confidence interval of specified confidence includes 0 or not.

Since, we are given the confidenceinterval, we will use that.

Given confidence interval is: b +/- 2 se = - 857 +/- (2 x 207) = - 857 +/- 414 = (- 1263, - 443). Since this interval does not hold 0, the relationship is significant.

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