Answer :
To solve this problem, we need to understand how the least squares regression equation works. The given equation is:
[tex]\[
\hat{y} = -173 + 74x
\][/tex]
Here's a step-by-step explanation:
1. Understanding the Equation:
- The regression equation is in the form [tex]\(\hat{y} = b_0 + b_1x\)[/tex], where:
- [tex]\(b_0\)[/tex] is the y-intercept, which in this case is [tex]\(-173\)[/tex].
- [tex]\(b_1\)[/tex] is the slope of the line, which in this case is [tex]\(74\)[/tex].
2. Interpretation of the Slope:
- The slope ([tex]\(b_1 = 74\)[/tex]) tells us how much [tex]\(\hat{y}\)[/tex] (the predicted value of [tex]\(y\)[/tex]) changes for a 1-unit increase in [tex]\(x\)[/tex].
- A positive slope means that as [tex]\(x\)[/tex] increases, [tex]\(\hat{y}\)[/tex] also increases.
3. Analyzing the Problem Statement:
- We need to figure out how [tex]\(\hat{y}\)[/tex] changes with a 1-unit increase in [tex]\(x\)[/tex].
- Since the slope is [tex]\(74\)[/tex], it indicates that [tex]\(\hat{y}\)[/tex] will increase by [tex]\(74\)[/tex] for each 1-unit increase in [tex]\(x\)[/tex].
Based on this analysis, each 1-unit increase in [tex]\(x\)[/tex] results in an increase in [tex]\(\hat{y}\)[/tex] by [tex]\(74\)[/tex].
Therefore, the correct choice is:
- A. increase; 74
[tex]\[
\hat{y} = -173 + 74x
\][/tex]
Here's a step-by-step explanation:
1. Understanding the Equation:
- The regression equation is in the form [tex]\(\hat{y} = b_0 + b_1x\)[/tex], where:
- [tex]\(b_0\)[/tex] is the y-intercept, which in this case is [tex]\(-173\)[/tex].
- [tex]\(b_1\)[/tex] is the slope of the line, which in this case is [tex]\(74\)[/tex].
2. Interpretation of the Slope:
- The slope ([tex]\(b_1 = 74\)[/tex]) tells us how much [tex]\(\hat{y}\)[/tex] (the predicted value of [tex]\(y\)[/tex]) changes for a 1-unit increase in [tex]\(x\)[/tex].
- A positive slope means that as [tex]\(x\)[/tex] increases, [tex]\(\hat{y}\)[/tex] also increases.
3. Analyzing the Problem Statement:
- We need to figure out how [tex]\(\hat{y}\)[/tex] changes with a 1-unit increase in [tex]\(x\)[/tex].
- Since the slope is [tex]\(74\)[/tex], it indicates that [tex]\(\hat{y}\)[/tex] will increase by [tex]\(74\)[/tex] for each 1-unit increase in [tex]\(x\)[/tex].
Based on this analysis, each 1-unit increase in [tex]\(x\)[/tex] results in an increase in [tex]\(\hat{y}\)[/tex] by [tex]\(74\)[/tex].
Therefore, the correct choice is:
- A. increase; 74
