Let's solve the problem step by step:
Given Data:
- Height (inches), [tex]\( x \)[/tex]: 27.75, 24.25, 25.75, 25.75, 24.75, 27.75, 26.25, 27.25, 26, 26, 27.5
- Head Circumference (inches), [tex]\( y \)[/tex]: 17.7, 17.3, 17.3, 17.7, 17.1, 17.8, 17.5, 17.7, 17.5, 17.7, 17.7
The linear regression equation is given by:
[tex]\[ \hat{y} = b_0 + b_1 x \][/tex]
Where:
- [tex]\( b_0 \)[/tex] is the y-intercept
- [tex]\( b_1 \)[/tex] is the slope
According to our calculations:
- Slope ([tex]\( b_1 \)[/tex]) = 0.1535073409461661
- Intercept ([tex]\( b_0 \)[/tex]) = 13.512398042414365
### (c) Predict the head circumference for a height of 24.75 inches
We use the regression equation to make this prediction. Substituting [tex]\( x = 24.75 \)[/tex]:
[tex]\[ \hat{y} = b_1 \cdot x + b_0 \][/tex]
[tex]\[ \hat{y} = 0.1535073409461661 \cdot 24.75 + 13.512398042414365 \][/tex]
[tex]\[ \hat{y} \approx 0.1535 \cdot 24.75 + 13.5124 \][/tex]
[tex]\[ \hat{y} \approx 3.799704 + 13.5124 \][/tex]
[tex]\[ \hat{y} \approx 17.31 \text{ in.} \][/tex]
Thus, the predicted head circumference for a child who is 24.75 inches tall is [tex]\( \hat{y} = 17.31 \)[/tex] inches.
### (d) Compute the residual
The residual is the difference between the observed head circumference and the predicted head circumference.
Observed head circumference for height of 24.75 inches is [tex]\( y = 17.1 \)[/tex].
The residual ([tex]\( \text{residual} \)[/tex]) is:
[tex]\[ \text{residual} = y - \hat{y} \][/tex]
[tex]\[ \text{residual} = 17.1 - 17.31 \][/tex]
[tex]\[ \text{residual} = -0.21 \][/tex]
Thus, the residual for the 24.75 inches tall child is [tex]\(-0.21\)[/tex]. This residual means that the head circumference of this child is [tex]\(0.21\)[/tex] inches below the value predicted by the regression model.
### Summary:
- (c) The predicted head circumference for a 24.75 inches tall child is [tex]\( \hat{y} = 17.31 \)[/tex] inches.
- (d) The residual for this observation is [tex]\(-0.21\)[/tex], meaning that the head circumference of this child is [tex]\(0.21\)[/tex] inches below the value predicted by the regression model.
