Answer :
To determine why the prediction is not accurate, let's consider the details given:
1. Regression Equation: The least squares regression line is given by [tex]\( Y = -1.11X + 68.17 \)[/tex], where [tex]\( X \)[/tex] is the weight of the cars in hundreds of pounds, and [tex]\( Y \)[/tex] is the miles per gallon (mpg) rating.
2. Weight Conversion: A car weighing 5000 pounds needs to be converted to hundreds of pounds for use in the regression equation:
[tex]\[
X = \frac{5000}{100} = 50
\][/tex]
3. Substitute and Predict: By substituting [tex]\( X = 50 \)[/tex] into the regression equation, you would calculate the predicted [tex]\( Y \)[/tex] (mpg):
[tex]\[
Y = -1.11 \times 50 + 68.17
\][/tex]
[tex]\[
Y = -55.5 + 68.17 = 12.67 \text{ mpg}
\][/tex]
4. Prediction Accuracy: The predicted value of 12.67 mpg is not close to the stated prediction of 67 mpg. Therefore, the prediction is inaccurate.
5. Reason for Inaccuracy:
- This is Considered Extrapolation: The given reason for inaccuracy is that this prediction for a car weighing 5000 pounds (50 in hundreds of pounds) is an extrapolation. Extrapolation happens when a prediction is made outside the range of the data set used to create the regression model. Extrapolations are less reliable because the behavior of the data outside the observed range may not continue in the same pattern.
Therefore, the correct reason why the prediction might not be accurate is that this is considered an extrapolation.
1. Regression Equation: The least squares regression line is given by [tex]\( Y = -1.11X + 68.17 \)[/tex], where [tex]\( X \)[/tex] is the weight of the cars in hundreds of pounds, and [tex]\( Y \)[/tex] is the miles per gallon (mpg) rating.
2. Weight Conversion: A car weighing 5000 pounds needs to be converted to hundreds of pounds for use in the regression equation:
[tex]\[
X = \frac{5000}{100} = 50
\][/tex]
3. Substitute and Predict: By substituting [tex]\( X = 50 \)[/tex] into the regression equation, you would calculate the predicted [tex]\( Y \)[/tex] (mpg):
[tex]\[
Y = -1.11 \times 50 + 68.17
\][/tex]
[tex]\[
Y = -55.5 + 68.17 = 12.67 \text{ mpg}
\][/tex]
4. Prediction Accuracy: The predicted value of 12.67 mpg is not close to the stated prediction of 67 mpg. Therefore, the prediction is inaccurate.
5. Reason for Inaccuracy:
- This is Considered Extrapolation: The given reason for inaccuracy is that this prediction for a car weighing 5000 pounds (50 in hundreds of pounds) is an extrapolation. Extrapolation happens when a prediction is made outside the range of the data set used to create the regression model. Extrapolations are less reliable because the behavior of the data outside the observed range may not continue in the same pattern.
Therefore, the correct reason why the prediction might not be accurate is that this is considered an extrapolation.
