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Wize AP Statistics Textbook > Linear Regression

Solving for the Regression Line

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Solving for the Regression Line (r, Sy, Sx Method)



Given a bunch of data coordinates (X,Y), we can generate a scatterplot. Then, we determine the best-fit line to represent the relationship between two quantitative variables: the explanatory variable XX and the response variable YY.

y^=bo+b1x\boxed{\hat{y}=b_o+b_1x}

The slope b1\colorFour{b_1}, tells us how much yy changes for every one unit increase in xx.

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Let’s prove this using calculus:

dy^dx=b1\displaystyle\boxed{\frac{d\hat{y}}{dx}=b_1}

“A unit increase in xx will change yy by b1b_1.”


Wize Concept
The slope b1b_1 and correlation coefficient rr always have the same sign!


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The intercept bo\colorFour{b_o}, tells us the value of yy when x=0x=0. It is the point where the line crosses the y-axis.



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Example: Solving for the Regression Line (r, Sy, Sx Method)


We want to see if there is a relationship between the number of hours a student studies the day before the exam and the exam grade. We randomly sample 34 students:



The explanatory variable (X) is:
Study (hours)

The response variable (Y) is:
Grade

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Scatterplot


We see a positive correlation. In fact, r=0.54r=0.54.

This means that there is a weak, positive correlation between hours studied and exam grade.

What does r2 tell us?

r2=(0.54)2=0.2916r^2=\left(0.54\right)^2=0.2916
About 29% of exam grade is explained by how many hours you study.


Portions of information contained in this publication/book are printed with permission of Minitab, LLC. All such material remains the exclusive property and copyright of Minitab, LLC. All rights reserved.

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Suppose you are given r, sx, sy, x‾, y‾r,\ s_x,\ s_y,\ \overline{x},\ \overline{y}:

x‾=7.794, y‾=77.824, sx=3.906, sy=18.235, r=0.54\overline{x}=7.794,\ \overline{y}=77.824,\ s_x=3.906,\ s_y=18.235,\ r=0.54

Step 1: Find the slope
b1=r(sysx)\displaystyle\boxed{b_1=r\left(\frac{s_y}{s_x}\right)}

b1=0.54(18.2353.906) =2.52\displaystyle{b_1=0.54\left(\frac{18.235}{3.906}\right)\ =2.52}



Step 2: Find the intercept
bo=y‾−b1x‾\displaystyle\boxed{b_o=\overline{y}-b_1\overline{x}}

bo=77.824−(2.52)(7.794)=58.18b_o=77.824-\left(2.52\right)\left(7.794\right)=58.18



Step 3: Show the full linear equation

y^=bo+b1x\displaystyle\boxed{\hat{y}=b_o+b_1x}
y^=58.18+2.52x\hat{y}=58.18+2.52x



Using the data set below, determine the correlation, slope, and intercept of the least squares regression line.



xˉ\bar{x}= 60 yˉ\bar{y}= 4.8

sxs_{x}= 38.08 sys_{y}= 2.59

r=???r=???

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Solving for the Regression Line (Least Squares Method)


Given a bunch of data coordinates (X,Y), we can generate a scatterplot. Then, we determine the best-fit line to represent the relationship between two quantitative variables: the explanatory variable XX and the response variable YY.

y^=bo+b1x\boxed{\hat{y}=b_o+b_1x}

The slope b1\colorFour{b_1}, tells us how much yy changes for every one unit increase in xx.

Let’s prove this using calculus:

dy^dx=b1\displaystyle\boxed{\frac{d\hat{y}}{dx}=b_1}

“A unit increase in xx will change yy by b1b_1.”

Wize Concept
The slope b1b_1 and correlation coefficient rr always have the same sign!

PAGE BREAK
The intercept bo\colorFour{b_o}, tells us the value of yy when x=0x=0. It is the point where the line crosses the y-axis.




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Example: Solving for the Regression Line (Sxy, Sxx Method)


We want to see if there is a relationship between the number of hours a student studies the day before the exam (XX) and the exam grade (YY). We randomly sample 34 students:

Note: raw data has been truncated


n=34n=34

Important: (∑xi)(∑yi)≠∑xiyi\left(\sum_{ }^{ }x_i\right)\left(\sum_{ }^{ }y_i\right)\neq\sum x_iy_i


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Scatterplot


Portions of information contained in this publication/book are printed with permission of Minitab, LLC. All such material remains the exclusive property and copyright of Minitab, LLC. All rights reserved.


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Step 1: Find the slope

b1=SSxySSxx\displaystyle\boxed{b_1=\frac{SS_{xy}}{SS_{xx}}}

SSxy=∑xiyi−(∑xi)(∑yi)n\displaystyle{SS_{xy}=\sum_{ }^{ }x_iy_i-\frac{\left(\sum_{ }^{ }x_i\right)\left(\sum_{ }^{ }y_i\right)}{n}}

SSxy=21,892−(265)(2,646)34=1,268.76\displaystyle{SS_{xy}=21,892-\frac{\left(265\right)\left(2,646\right)}{34}=1,268.76}


SSxx=∑xi2−(∑xi)2n\displaystyle{SS_{xx}=\sum_{ }^{ }x_i^2-\frac{\left(\sum_{ }^{ }x_i\right)^2}{n}}


SSxx=2,569−(265)234=503.56\displaystyle{SS_{xx}=2,569-\frac{\left(265\right)^2}{34}=503.56}

Therefore:

b1=SSxySSxx=1,268.76503.56=2.52\displaystyle{b_1=\frac{SS_{xy}}{SS_{xx}}=\frac{1,268.76}{503.56}=2.52}


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Step 2: Find the intercept
bo=y‾−b1x‾\displaystyle\boxed{b_o=\overline{y}-b_1\overline{x}}

y‾=∑yin\displaystyle{\overline{y}=\frac{\sum_{ }^{ }y_i}{n}}

y‾=2,64634=77.824\displaystyle{\overline{y}=\frac{2,646}{34}=77.824}

x‾=∑xin\displaystyle{\overline{x}=\frac{\sum_{ }^{ }x_i}{n}}

x‾=26534=7.794\overline{x}=\frac{265}{34}=7.794

Therefore:

bo=77.824−(2.52)(7.794)=58.18b_o=77.824-\left(2.52\right)\left(7.794\right)=58.18


Step 3: Show the full linear equation

y^=bo+b1x\boxed{\hat{y}=b_o+b_1x}
y^=58.18+2.52x\hat{y}=58.18+2.52x
Using the information provided below, solve for the slope and intercept of the linear regression equation.

∑x=702∑y=461∑xy=19,871∑x2=31396∑y2=13,449\begin{array}{ll}\sum x=702\\\sum y=461\\\sum xy=19,871\\\sum x^2=31396\\\sum y^2=13,449\end{array}
n=18n=18
Click on 'HINT' if you are stuck!