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

The correlation coefficient \( r = 0.003 \) is very close to 0, indicating a very weak linear relationship between the variables.

Step2: Determine Prediction Method

When the linear correlation is weak ( \( |r| \) is close to 0), the best predicted value of the response variable \( y \) for a given \( x \) is the mean of the \( y \)-values, \( \bar{y} \). But wait, wait—no, wait. Wait, actually, wait: the regression equation is given, but if correlation is weak, but maybe I made a mistake. Wait, no, let's recheck. Wait, the regression equation is \( \hat{y}=2 + 3x \). Wait, maybe the key is: when \( r \) is close to 0, but the regression equation is given. Wait, no, the rule is: if the linear correlation coefficient \( r \) indicates a weak linear relationship ( \( |r| \) is small), then the best predicted value of \( y \) for a given \( x \) is the mean of \( y \), \( \bar{y} \). But wait, the problem says "ten pairs of data yield \( r = 0.003 \)" which is very close to 0, so the linear relationship is almost non - existent. But wait, the regression equation is \( \hat{y}=2 + 3x \). Wait, maybe I confused the rule. Wait, no, let's think again. The formula for the predicted value: if we use the regression equation, when \( x = 2 \), \( \hat{y}=2+3\times2=8 \). But wait, the correlation is weak. Wait, no, the rule is: if there is a linear correlation (even weak), but when \( r \) is close to 0, the best prediction is the mean? Wait, no, maybe the question is a trick. Wait, no, let's calculate using the regression equation first. Let's compute \( \hat{y} \) when \( x = 2 \).

Step3: Calculate using Regression Equation

Given the regression equation \( \hat{y}=2 + 3x \). Substitute \( x = 2 \) into the equation:
\( \hat{y}=2+3\times2=2 + 6=8 \)
Wait, but the correlation is weak. But maybe the question is designed to use the regression equation regardless? Wait, let's check the options. The mean \( \bar{y}=5.0 \), but option B is 5.0, but when we calculate using the regression equation, we get 8.0. Wait, maybe my initial thought about the rule was wrong. Let's recall: the regression equation is used to predict \( y \) when there is a linear relationship. If the correlation is weak, but the regression equation is given, maybe we still use the regression equation? Wait, the value of \( r = 0.003 \) is very close to 0, which means that the linear model is not a good fit. But in some cases, if the problem gives the regression equation, maybe we are supposed to use it. Let's recalculate: \( \hat{y}=2+3x \), \( x = 2 \), so \( \hat{y}=2 + 3\times2=8 \). So the answer should be 8.0.

Answer:

C. 8.0