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A pediatrician wants to determine the relation that exists between a child's height, [tex]$x$[/tex], and head circumference, [tex]$y$[/tex]. She randomly selects 11 children from her practice, measures their heights and head circumferences, and obtains the accompanying data. Complete parts (a) through (g) below.

Click the icon to view the children's data.

(c) Use the regression equation to predict the head circumference of a child who is 24.75 inches tall.
[tex]$
\hat{ y }=\square \text { in. }
$[/tex]
(Round to two decimal places as needed.)

(d) Compute the residual based on the observed head circumference of the 24.75-inch-tall child in the table. The residual for this observation is [tex]$\square$[/tex], meaning that the head circumference of this child is [tex]$\square$[/tex] above the value predicted.

(e) Draw the least-squares regression line on the scatter diagram of the data and label the residual from part (d).

Data Table
\begin{tabular}{|c|c|}
\hline
Height (inches), [tex]$x$[/tex] & Head Circumference (inches), [tex]$y$[/tex] \\
\hline
27.75 & 17.7 \\
24.25 & 17.3 \\
25.75 & 17.3 \\
25.75 & 17.7 \\
24.75 & 17.1 \\
27.75 & 17.8 \\
26.25 & 17.5 \\
27.25 & 17.7 \\
26 & 17.5 \\
26 & 17.7 \\
27.5 & 17.7 \\
\hline
\end{tabular}



Answer :

GinnyAnswer
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.