Question

use the given data to find the best predicted value of the response variable. ten pairs of data yield r = 0.003 and the regression equation (hat{y} = 2 + 3x). also, (\bar{y} = 5.0). what is the best predicted value of (y) for (x = 2)?

a. 7.0
b. 5.0
c. 8.0
d. 17.0

Explanation:

Step1: Analyze correlation coefficient

The correlation coefficient \( r = 0.003 \) is very close to 0, indicating a very weak linear relationship. However, when predicting the response variable, if the correlation is weak, but we have a regression equation, we still use the regression equation for prediction (or sometimes the mean, but here the regression equation is given and we check the logic). Wait, actually, when \( r \) is close to 0, the best predicted value might be the mean \( \bar{y} \)? Wait no, wait the regression equation is \( \hat{y}=2 + 3x \). Wait, let's check the steps again.

Wait, the problem is to find the best predicted value of \( y \) for \( x = 2 \). The regression equation is \( \hat{y}=2+3x \). Let's substitute \( x = 2 \) into the regression equation.

Step2: Substitute \( x = 2 \) into regression equation

Substitute \( x = 2 \) into \( \hat{y}=2 + 3x \):
\( \hat{y}=2+3(2)=2 + 6 = 8 \). Wait, but wait the correlation coefficient is 0.003, which is very low. But maybe the question is using the regression equation regardless? Wait, but let's check the options. Wait, maybe I made a mistake. Wait, the mean \( \bar{y}=5.0 \), but when the correlation is weak, do we use the mean or the regression? Wait, no—actually, the regression equation is used for prediction, even with weak correlation, unless it's stated otherwise. Wait, but let's recalculate: \( 2 + 3(2)=8 \), which is option C. But wait, maybe the question has a typo? Wait, no, let's check again.

Wait, maybe the initial thought was wrong. Let's do the substitution again: \( x = 2 \), so \( 2 + 3*2 = 2 + 6 = 8 \). So the best predicted value using the regression equation is 8.0.

Answer:

C. 8.0