Question

Heights of fathers and sons are given in centimeters.
Height of father
150
152
155
157
160
161
164
166
Height of son
154
156
158
159
160
162
161
164
Find the line of regression and calculate the expected average height of the son when the height of the father is 154cm.

Answer 1

Answer on Question #43725-Math-Statistics and Probability

Heights of fathers and sons are given in centimeters.

Height of father (x)

150 152 155 157 160 161 164 166

Height of son (y)

154 156 158 159 160 162 161 164

Find the line of regression and calculate the expected average height of the son when the height of the father is 154cm.

Solution

Let 160 and 159 be assumed means of $x$ and $y$. Using the given data, we get the following table:

$$\bar{x} = 160 + \frac{\sum X}{n} = 160 - \frac{15}{8} = 158.13$$ $$\bar{y} = 159 + \frac{\sum Y}{n} = 159 + \frac{2}{8} = 159.25$$

Since regression coefficients are independent of change of origin, we have regression coefficient of $y$ on $x$.

$$b_{yx} = b_{YX} = \frac{n \sum XY - \sum X \sum Y}{n \sum X^2 - (\sum X)^2} = \frac{8 \cdot 120 - (-15) \cdot 2}{8 \cdot 251 - (-15)^2} = 0.56$$

Equation of a line of regression of $y$ on $x$ is

$$y - \bar{y} = b_{yx}(x - \bar{x})$$ $$y - 159.25 = 0.56(x - 158.13)$$ $$y = 0.56x + 70.697.$$

When $x = 154$

$$y = 0.56(154) + 70.697 = 156.937.$$

Answer: $y = 0.56x + 70.697; y = 156.937$.

www.AssignmentExpert.com