Question

Because elderly people may have difficulty standing to have their height measured, a study looked at the relationship between overall height and height to the knee. Here are data (in centimeters) for five elderly men:

col1 Knee Height x 56 44 41 44 55
col2 Height y 190 150 145 165 172
What is the equation of the least-squares regression line for predicting height from knee height?

Answer 1

Answer:

The equation of the least-squares regression line for predicting height from knee height is $y~=~53.307 ~+~ 2.314 \cdot x$.

Step-by-step explanation:

Given any collection of pairs of numbers, there always exists exactly one straight line that fits the data better than any other, in the sense of minimizing the sum of the squared errors. It is called the least squares regression line.

The slope-intercept form of a line is

                                                $y = bx + a$

where b is the slope of the line and a is the y-intercept.

We have the following data

$\begin{array}{cc|ccccc}Knee \:Height&x&56&44&41&44&55\\Height&y&190&150&145&165&172\end{array}$

To find the equation of the regression line you must:

Step 1: Find $x\cdot y$ and $x^2$ as it was done in the table below.

Step 2: Find the sum of every column.

$\sum{X} = 240 ~,~ \sum{Y} = 822 ~,~ \sum{X \cdot Y} = 39905 ~,~ \sum{X^2} = 11714$

Step 3: Use the following equations to find a and b

$\begin{aligned} a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 822 \cdot 11714 - 240 \cdot 39905}{ 5 \cdot 11714 - 240^2} \approx 53.307 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 5 \cdot 39905 - 240 \cdot 822 }{ 5 \cdot 11714 - \left( 240 \right)^2} \approx 2.314\end{aligned}$

Step 4: Substitute a and b in the regression equation formula

$\begin{aligned} y~&=~a ~+~ b \cdot x \\y~&=~53.307 ~+~ 2.314 \cdot x\end{aligned}$