Question

The Heights(in inches)and the weight (in pounds)of a simplified women are recorded. round your answer to one decimal place ×=height. y=weight 66. 122 67. 13269. 15368. 13865. 125a) calculate the slope of B1 and the y-intercept Bl0) of the regression line. Write the regression equation.y= _____ x+_____b) Interest interpret the values of B1 and B0. For it each increase of 1 inches the estimate weight increase by_______ pounds. When x=0 inches weight the estimate weight is________ pounds

Answer 1

For the linear regression the dependent and independent variables are:

y: "weight of a woman" (measured in pounds)

x: "height of a woman" (measured in inches)

You have to calculate the estimated linear regression equation:

$y^{}=b_{0\text{ }}+b_1x$

Where

b₀ is the estimate of the y-intercept

b₁ is the estimate of the slope

To calculate both by hand you have to use the following formulas:

$b_1=\frac{\sum ^n_1x_iy_i-\frac{(\sum^n_1x_i)(\sum^n_1y_i)}{n}}{\sum ^n_1x^2_i-\frac{(\sum ^n_1x_i)^2}{n}}$$b_0=y_{\text{bar}}+bx_{\text{bar}}$

y bar and x bar are the sample means for the weight and height

n is the number of women observed, in this case the sample size is

n=5 women

Next calculate the needed summatories, i.e. add each obervation from the corresponding variables

∑xi= 335

∑xi²= 22455

∑yi=670

∑xiyi=44962

ybar=∑yi/n= 670/5=134 pounds

xbar=∑xi/n= 335/5=67 inches

Replace the values in the formulas:

Slope:

$b_1=\frac{44962-\frac{335\cdot670}{5}}{22455-\frac{(335)^2}{5}}=7.20\text{ }\frac{pounds}{In}$

y-intercept

$b_0=134-7.20\cdot67=-348.4\text{pounds}$

The estimated linear equation is:

y=-348.4+7.20x

b)

The y-intercept of the linear regression indicates the value of the estimated mean of the variable y when x equals zero.

The slope of the linear regresion insicates the change of the estimated mean of the variable y when the variable x increases one unit.

So in terms of this exercise you can interpret the y-intercept as:

-348.4pounds is the estimated average weight of women when their height is zero inches.

→ Note that this interpretation has no logical meaning, at least not one applicable in real life, but for the regression is a valid mathematical result and interpretation.

And the slope can be interpreted as:

7.2pounds/inches is the modification of the estimated average weight of weomen when their height increases 1 inch.