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Linear Regression
Related Topics
Wize University Statistics Textbook > Simple Linear Regression
Solving for the Regression Line
6 Activities
x
ˉ
\bar{x}
x
ˉ
= 60
y
ˉ
\bar{y}
y
ˉ
= 4.8
s
x
s_{x}
s
x
= 38.08
s
y
s_{y}
s
y
= 2.59
r
=
0.71
r=0.71
r
=
0.71
Part 1
Part 2
A) Determine the equation of the least squares regression line
Step 1: Find the slope
b
=
r
s
y
s
x
b=r\frac{s_{y}}{s_{x}}
b
=
r
s
x
s
y
Step 2: Find the intercept
a
=
y
ˉ
−
b
x
ˉ
a=\bar{y}-b\bar{x}
a
=
y
ˉ
−
b
x
ˉ
Slope
b
=
b=
b
=
Intercept
a
=
a=
a
=
Equation of the line
y
^
=
\hat y=
y
^
=
I don't know
Previous Part
Check Part 1
Next Part
Check Submission
More Solving for the Regression Line Questions:
Linear Regression
x
ˉ
\bar{x}
x
ˉ
= 60
y
ˉ
\bar{y}
y
ˉ
= 4.8
s
x
s_{x}
s
x
= 38.08
s
y
s_{y}
s
y
= 2.59
r
=
0.71
r=0.71
r
=
0.71
Regression line and Residual plots
Using the data set below, determine the correlation, slope, and intercept of the least squares regression line.
x
ˉ
\bar{x}
x
ˉ
= 60
y
ˉ
\bar{y}
y
ˉ
= 4.8
s
x
s_{x}
s
x
= 38.08
s
y
s_{y}
s
y
= 2.59
Practice (Version 1)
Using the data set below, determine the correlation and intercept of the least squares regression line.
∑
x
=
300
\sum{x}=300
∑
x
=
300
∑
y
=
24
\sum{y}=24
∑
y
=
24
s
x
=
38.08
s_{x}=38.08
s
x
=
38.08
s
y
=
2.59
s_{y}=2.59
s
y
=
2.59
Correlation and Regression Line
If
r
=
−
1.00
r=-1.00
r
=
−
1.00
, then 100% of the data points fall exactly on the regression line and the slope cannot be equal or greater than 0.
Regression Line
The following data is obtained to assess the relationship between height in cm and shoe size.
The correlation coefficient is 0.93
Which of the following is the equation of the regression line?
Regression Line
One day George wanted to examine the linear relationship between how many hours a student studies (
x
x
x
) the day before the exam and the exam grade (
y
y
y
).
He asks 13 random students and collects the following data
r
=
0.819
s
x
=
3.24
s
y
=
14.24
x
‾
=
8.25
y
‾
=
78.25
r=0.819\ \ s_x=3.24\ \ s_y=14.24\ \ \overline{x}=8.25\ \ \overline{y}=78.25
r
=
0.819
s
x
=
3.24
s
y
=
14.24
x
=
8.25
y
=
78.25
Regression Line
In investigating the relationship between number of people on a person's phone list (
x
x
x
) and the number of text messages per day (
y
y
y
), 40 individuals were surveyed and the following information was recorded:
∑
i
=
1
40
x
i
=
700
s
x
=
3.5
\sum_{i=1}^{40}x_i=700 \quad s_x=3.5
i
=
1
∑
40
x
i
=
700
s
x
=
3.5
∑
i
=
1
40
y
i
=
3200
s
y
=
18.9
r
=
0.67
\sum_{i=1}^{40}y_i=3200 \quad s_y=18.9 \quad r=0.67
i
=
1
∑
40
y
i
=
3200
s
y
=
18.9
r
=
0.67
Regression Line
The equation
y
^
=
3
+
0.025
x
\hat y=3+0.025x
y
^
=
3
+
0.025
x
is used to predict the amount of weight lost
y
y
y
(in kg) by a group of individuals in a study based on the amount of their daily exercise
x
x
x
(in minutes). Given that
1
k
g
≈
2.2
l
b
1kg\approx2.2lb
1
k
g
≈
2.2
l
b
, suppose we want to change the amount of weight lost to pounds (lb) in this equation, which of the following will be the correct linear regression line?
Slope of a Regression Line
Suppose we want to see if there is a relationship between the number of hours a student studies the day before their exam (x) and their exam grade (y). We randomly sample 8 students and record our results, the correlation coefficient is 0.819.
If a student studies an additional 2 hours the day before their final exam, by how much can they expect their mark to increase?
Linear Regression
Here's the data and summary of the weekly earnings and amount of coffee purchased by 5 different students:
x
ˉ
\bar{x}
x
ˉ
= 60
y
ˉ
\bar{y}
y
ˉ
= 4.8
s
x
s_{x}
s
x
= 38.08
s
y
s_{y}
s
y
= 2.59
r
=
0.71
r=0.71
r
=
0.71
Regression Line: Changing Units of $x$
The equation
y
^
=
35
+
1.50
x
\hat y=35+1.50x
y
^
=
35
+
1.50
x
is used to predict the hourly wage y of someone with x years of experience.
Suppose that x is changed to months of experience. Which of the following will be the correct regression line?
Regression line
Suppose we want to see if there is a relationship between the size of a condo unit and the selling price of it. We randomly sampled 32 units:
r
=
0.762
s
x
=
278.87
s
y
=
60350.59
x
ˉ
=
1034.84
y
ˉ
=
445734.38
\begin{array}{c}r=0.762\\s_x=278.87\\s_y=60350.59\\\bar{x}=1034.84\\\bar{y}=445734.38\end{array}
r
=
0.762
s
x
=
278.87
s
y
=
60350.59
x
ˉ
=
1034.84
y
ˉ
=
445734.38
With this linear regression plot
Regression Line
Suppose we want to see if there is a relationship between the size of a condo unit and the selling price of it. We randomly sampled 32 units:
r
=
0.762
s
x
=
278.87
s
y
=
60350.59
x
ˉ
=
1034.84
y
ˉ
=
445734.38
\begin{array}{c}r=0.762\\s_x=278.87\\s_y=60350.59\\\bar{x}=1034.84\\\bar{y}=445734.38\end{array}
r
=
0.762
s
x
=
278.87
s
y
=
60350.59
x
ˉ
=
1034.84
y
ˉ
=
445734.38
With this linear regression plot
Regression Line
Suppose we want to see if there is a relationship between the size of a condo unit and the selling price of it. We randomly sampled 32 units:
r
=
0.762
s
x
=
278.87
s
y
=
60350.59
x
ˉ
=
1034.84
y
ˉ
=
445734.38
\begin{array}{c}r=0.762\\s_x=278.87\\s_y=60350.59\\\bar{x}=1034.84\\\bar{y}=445734.38\end{array}
r
=
0.762
s
x
=
278.87
s
y
=
60350.59
x
ˉ
=
1034.84
y
ˉ
=
445734.38
With this linear regression plot
Regression Line: Calculation
One day George wanted to examine the linear relationship between how many hours a student studies (
x
x
x
) the day before the exam and the exam grade (
y
y
y
).
He asks 13 random students and collects the following data
r
=
0.819
s
x
=
3.24
s
y
=
14.24
x
‾
=
8.25
y
‾
=
78.25
r=0.819\ \quad s_x=3.24\ \quad s_y=14.24\ \quad\overline{x}=8.25\ \ \overline{y}=78.25
r
=
0.819
s
x
=
3.24
s
y
=
14.24
x
=
8.25
y
=
78.25
Regression Line: Challenging Use of Slope
Suppose we want to see if there is a relationship between the number of hours a student studies the day before their exam (x) and their exam grade (y). We randomly sample 8 students and record our results. The correlation coefficient is 0.819.
If a student studies an additional 2 hours the day before their final exam, by how much can they expect their mark to increase?
Regression Line: Changing Units of X
The equation
y
^
\hat{y}
y
^
=35+1.50
x
x
x
is used to predict the hourly wage y of someone with x years of experience. Suppose that x is changed to months of experience. Which of the following will be the correct regression line?
Regression Line: Changing Units of Y Example
Problem
EThe equation
w
e
i
g
h
t
L
o
s
t
^
\widehat{weight Lost}
w
e
i
g
h
t
L
os
t
=3+0.025 (
e
x
e
r
c
i
s
e
^
\widehat{exercise}
e
x
er
c
i
se
) is used to predict the amount of weight lost, in kilograms, by a group of individuals in a study based on the average amount of daily exercise, in minutes. Each kilogram is approximately 2.2 pounds. Suppose that we change weight lost to pounds. Which of the following describes the corresponding linear regression line?
Regression Line
Refer to Question 19.
What is the equation of the regression line?
Regression line and Residual plots
Using the data set below, determine the correlation, slope, and intercept of the least squares regression line.
x
ˉ
\bar{x}
x
ˉ
= 60
y
ˉ
\bar{y}
y
ˉ
= 4.8
s
x
s_{x}
s
x
= 38.08
s
y
s_{y}
s
y
= 2.59
Solving for the Regression Line
The equation
y
^
=
3
+
0.025
x
\hat{y}=3+0.025x
y
^
=
3
+
0.025
x
is used to predict the amount of weight lost
y
y
y
, in kilograms, by a group of individuals in a study based on the average amount of daily exercise
x
x
x
, in minutes. Each kilogram is approximately 2.2 pounds.
Suppose that we change weight lost to pounds. Determine the new linear regression line.
Regression Line
What is the intercept of the regression line?
Regression Line
Suppose Big Joe is added to the scatterplot below. He is 13 years old and has 50% body fat. What will happen to the slope and correlation coefficient?
Correlation and Regression Line
Suppose we want to see if there is a relationship between the number of times a student attended their labs in the entire semester (x) and their final exam grade (y). We randomly sample 8 students and recorded our results.
∑
(
x
i
−
x
‾
)
(
y
i
−
y
‾
)
=
264.6
\sum_{ }^{ }\left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)=264.6
∑
(
x
i
−
x
)
(
y
i
−
y
)
=
264.6
Regression line: Predictions and Residual Plots
True or false?
If the sum of the residuals is equal to 0, then the best-fitting regression line goes through all the data points.
