Q. 10 - The Practice of Statistics for AP

Question

Social scientists are interested in the association between the high school graduation rate (HSGR) and the percentage of U.S. families living in poverty (POV). Data were collected from all $50$ states and the District of Columbia, and a regression analysis was conducted. The resulting least-squares regression line is given by $\overline{POV}=59.2-0.620$(HSGR) with $r^2=0.802$. Based on the information, which of the following is the best interpretation for the slope of the least-squares regression line?

(a) For each $1\%$ increase in the graduation rate, the per cent of families living in poverty is predicted to decrease by approximately $0.896$.

(b) For each $1\%$ increase in the graduation rate, the per cent of families living in poverty is predicted to decrease by approximately $0.802$.

(c) For each $1\%$ increase in the graduation rate, the per cent of families living in poverty is predicted to decrease by approximately $0.620$.

(d) For each$1\%$ increase in the percentage of families living in poverty, the graduation rate is predicted to increase by approximately $0.802$.

(e) For each $1\%$ increase in the per cent of families living in poverty, the graduation rate is predicted to decrease by approximately $0.620$. 

Step-by-Step Solution

Answer

The best interpretation for the slope of the least-squares regression line is option (c) For each $1\%$ increase in the graduation rate, the per cent of families living in poverty is predicted to decrease by approximately $0.620.$

Step 1: Given information

Data were collected from all $50$ states

The least-squares regression line is given by $\overline{POV}=59.2-0.620$(HSGR) with $r^2=0.802$ 

To find the best interpretation for the slope of the least-squares regression line.

Step 2: Explanation

Determine the regression line's slope; observe that it is negative, indicating a negative association between high school graduation rate and poverty (as graduation rate increases, poverty decreases; graduation rate decreases, poverty increases)

Slope $=-0.620$

Notice answers a, b, and d are not using the value of the slope. This leaves only two answers

Remove option e

The slope is calculated as follows: for every unit increase in X, the slope value of 0.620 decreases Y. (decreased since negative). In this scenario, X represents the high school graduation rate and Y represents the poverty rate. Part e is a mash-up, and the correct answer is (c).