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An old study done in the 1920s considered the relationship between a farm's corn yield in bushels per acre and the land value per acre of that farm. The study found that the data had a least squares regression equation of [tex]\hat{y}=12.32x-273.6[/tex]. Suppose that one of these farms produced 35 bushels of corn per acre and had a land value of [tex]\$120[/tex] per acre. Calculate the residual for this data point.

A) -153.6
B) 158
C) 37.6
D) -37.6

Answer :

To solve the problem of finding the residual, you need to use the given least squares regression equation:

[tex]\[
\hat{y} = 12.32x - 273.6
\][/tex]

Here, [tex]\( x \)[/tex] represents the number of bushels of corn produced per acre. For the farm described, [tex]\( x = 35 \)[/tex] bushels per acre. The actual land value for this farm is [tex]$120 per acre.

Step 1: Calculate the Predicted Land Value (\(\hat{y}\))

First, substitute \( x = 35 \) into the regression equation to find the predicted land value:

\[
\hat{y} = 12.32 \times 35 - 273.6
\]

Calculate the result:

1. Multiply 12.32 by 35:
\[
12.32 \times 35 = 431.2
\]

2. Subtract 273.6 from 431.2 to find the predicted land value:
\[
431.2 - 273.6 = 157.6
\]

So, the predicted land value is approximately $[/tex]157.6 per acre.

Step 2: Calculate the Residual

The residual is the difference between the actual land value and the predicted land value. So, you use the formula:

[tex]\[
\text{Residual} = \text{Actual value} - \text{Predicted value}
\][/tex]

Substitute the given values:

[tex]\[
\text{Residual} = 120 - 157.6 = -37.6
\][/tex]

Conclusion

The residual for this data point is -37.6. Therefore, the correct answer is option D) -37.6.

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