Question

a researcher is studying the relationship between fathers and sons heights. he collects a simple random sample of eight pairs of fathers and sons and records their heights as shown in the table.
the equation of the least - squares regression line is \\( \hat { y } = 2.7 + 1.042 x \\), where \\( \hat { y } \\) is each sons height and \\( x \\) is his fathers height. which shows the residual plot?

Explanation:

Step1: Calculate residuals

Residual formula: \(e = y-\hat{y}\), where \(y\) is the observed value and \(\hat{y}=2.7 + 1.042x\) is the predicted value.
For \(x = 74\): \(\hat{y}=2.7+1.042\times74=2.7 + 77.108=79.808\), \(e=77 - 79.808=- 2.808\)
For \(x = 67\): \(\hat{y}=2.7+1.042\times67=2.7+69.814 = 72.514\), \(e=70 - 72.514=-2.514\)
For \(x = 61\): \(\hat{y}=2.7+1.042\times61=2.7+63.562 = 66.262\), \(e=67 - 66.262 = 0.738\)
For \(x = 75\): \(\hat{y}=2.7+1.042\times75=2.7+78.15=80.85\), \(e=85 - 80.85 = 4.15\)
For \(x = 60\): \(\hat{y}=2.7+1.042\times60=2.7+62.52=65.22\), \(e=66 - 65.22 = 0.78\)
For \(x = 63\): \(\hat{y}=2.7+1.042\times63=2.7+65.646 = 68.346\), \(e=67 - 68.346=-1.346\)
For \(x = 63\): \(\hat{y}=2.7+1.042\times63=2.7+65.646 = 68.346\), \(e=71 - 68.346 = 2.654\)
For \(x = 66\): \(\hat{y}=2.7+1.042\times66=2.7+68.772 = 71.472\), \(e=70 - 71.472=-1.472\)

Step2: Analyze the residual plot

The \(x\) - axis is father's height (\(x\)) and \(y\) - axis is residual (\(e\)). Plot the points \((74,-2.808)\), \((67,-2.514)\), \((61,0.738)\), \((75,4.15)\), \((60,0.78)\), \((63,-1.346)\), \((63,2.654)\), \((66,-1.472)\)

Answer:

The residual plot with points corresponding to the calculated residuals (as per the \(x\) (father's height) and \(y\) (residual) values) is the correct one.