Question

use the model to predict the mean weight of boys age 5.
the actual mean weight of boys age 5 is 40. determine the residual value for boys age 5. does this value indicate an overestimate or underestimate? explain.

Explanation:

Step1: Find the linear regression equation

Let \(x\) be the age and \(y\) be the mean weight. Using the formula for the slope \(m=\frac{n\sum(xy)-\sum x\sum y}{n\sum(x^{2})-(\sum x)^{2}}\) and the intercept \(b = \frac{\sum y - m\sum x}{n}\), where \(n = 6\) (number of data points).

\(\sum x=2 + 4+6 + 8+10 + 12=42\), \(\sum y=27.5+36 + 45.5+56.5+70.5+88 = 324\)

\(\sum(xy)=2\times27.5+4\times36+6\times45.5+8\times56.5+10\times70.5+12\times88\)
\(=55+144+273+452+705+1056=2685\)

\(\sum(x^{2})=2^{2}+4^{2}+6^{2}+8^{2}+10^{2}+12^{2}=4 + 16+36+64+100+144 = 364\)

\(m=\frac{6\times2685-42\times324}{6\times364-(42)^{2}}=\frac{16110 - 13608}{2184 - 1764}=\frac{2502}{420}=5.9571\approx6\)

\(b=\frac{324-6\times42}{6}=\frac{324 - 252}{6}=\frac{72}{6}=12\)

The linear regression equation is \(y = 6x+12\)

Step2: Predict the weight for \(x = 5\)

Substitute \(x = 5\) into \(y=6x + 12\), we get \(y=6\times5+12=30 + 12=42\)

Step3: Calculate the residual

The residual \(r=\text{Actual value}-\text{Predicted value}\)
Given actual value \(y_{actual}=40\), predicted value \(y_{predicted}=42\)
\(r=40 - 42=- 2\)

Answer:

The residual value is \(-2\). Since the residual is negative (\(r=-2<0\)), the predicted value (\(42\)) overestimates the actual value (\(40\)).