Questions: For 39 nations, a correlation of 0.863 was found between y= Internet use (%) and x= gross domestic product (GDP, in thousands of dollars per capita). The regression equation is y^=-3.37+1.53 x. Complete parts (a) through (c). a. Based on the correlation value, the slope had to be positive. Why? A. That is a very unusual fact, because the slope and correlation usually have different signs. B. Although slope and correlation usually have different values, they always have the same sign. C. The slope and correlation are positive because gross domestic product could not be negative. D. The correlation and the slope are positive because the y-intercept is negative. b. One nation had a GDP of 25.6 thousand dollars and Internet use of 30.6 %. Find its predicted Internet use based on the regression equation. % (Round to one decimal place as needed.) c. Find the residual for this nation. (Round to one decimal place as needed.)

For 39 nations, a correlation of 0.863 was found between y= Internet use (%) and x= gross domestic product (GDP, in thousands of dollars per capita). The regression equation is y^=-3.37+1.53 x. Complete parts (a) through (c).
a. Based on the correlation value, the slope had to be positive. Why?
A. That is a very unusual fact, because the slope and correlation usually have different signs.
B. Although slope and correlation usually have different values, they always have the same sign.
C. The slope and correlation are positive because gross domestic product could not be negative.
D. The correlation and the slope are positive because the y-intercept is negative.
b. One nation had a GDP of 25.6 thousand dollars and Internet use of 30.6 %. Find its predicted Internet use based on the regression equation.
% (Round to one decimal place as needed.)
c. Find the residual for this nation.
(Round to one decimal place as needed.)

Transcript text: For 39 nations, a correlation of 0.863 was found between $y=$ Internet use (\%) and $x=$ gross domestic product (GDP, in thousands of dollars per capita). The regression equation is $\hat{y}=-3.37+1.53 x$. Complete parts (a) through (c). a. Based on the correlation value, the slope had to be positive. Why? A. That is a very unusual fact, because the slope and correlation usually have different signs. B. Although slope and correlation usually have different values, they always have the same sign. C. The slope and correlation are positive because gross domestic product could not be negative. D. The correlation and the slope are positive because the $y$-intercept is negative. b. One nation had a GDP of 25.6 thousand dollars and Internet use of $30.6 \%$. Find its predicted Internet use based on the regression equation. $\%$ (Round to one decimal place as needed.) c. Find the residual for this nation. (Round to one decimal place as needed.)

Solution

Solution Steps

Step 1: Correlation and Slope Relationship

The correlation coefficient $ r = 0.863 $ indicates a strong positive relationship between Internet use ($ y $) and GDP ($ x $). The slope of the regression line is $ \beta = 1.53 $, which is also positive. This confirms that the slope and correlation always have the same sign. Therefore, the correct answer to the multiple-choice question is:

\[ \text{Answer: B. Although slope and correlation usually have different values, they always have the same sign.} \]

Step 2: Predicted Internet Use Calculation

Using the regression equation $ \hat{y} = -3.37 + 1.53x $, we can calculate the predicted Internet use for a nation with a GDP of $ 25.6 $ thousand dollars:

\[ \hat{y} = -3.37 + 1.53 \times 25.6 \]

Calculating this gives:

\[ \hat{y} = -3.37 + 39.148 = 35.778 \]

Rounding to one decimal place, the predicted Internet use is:

\[ \hat{y} \approx 35.8\% \]

Step 3: Residual Calculation

The residual is calculated as the difference between the actual Internet use and the predicted Internet use:

\[ \text{Residual} = \text{Actual Internet Use} - \text{Predicted Internet Use} \]

Substituting the values:

\[ \text{Residual} = 30.6 - 35.778 = -5.178 \]

Rounding to one decimal place, the residual is:

\[ \text{Residual} \approx -5.2\% \]