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Correlation and Regression Line

Related Topics

Wize University Statistics Textbook > Simple Linear Regression

Solving for the Regression Line

6 Activities

Wize University Statistics Textbook > Scatterplots, Association, and Correlation

Solving for Correlation

5 Activities

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.


∑(xi−x‾)(yi−y‾)=264.6\sum_{ }^{ }\left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)=264.6
r=0.819r=0.819

If a student attends an additional 2 labs before their final exam, by how much can they expect their grade to increase?

[This question is hard. Click on 'HINT' if you are stuck.]
More Solving for the Regression Line Questions:
Linear Regression
xˉ\bar{x}= 60 yˉ\bar{y}= 4.8
sxs_{x}= 38.08 sys_{y}= 2.59 r=0.71r=0.71
Regression line and Residual plots
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
Practice (Version 1)
Using the data set below, determine the correlation and intercept of the least squares regression line.
∑x=300\sum{x}=300 ∑y=24\sum{y}=24
sx=38.08s_{x}=38.08 sy=2.59s_{y}=2.59
Correlation and Regression Line
If r=−1.00r=-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 (xx) the day before the exam and the exam grade (yy).
He asks 13 random students and collects the following data
r=0.819  sx=3.24  sy=14.24  x‾=8.25  y‾=78.25r=0.819\ \ s_x=3.24\ \ s_y=14.24\ \ \overline{x}=8.25\ \ \overline{y}=78.25
Regression Line
In investigating the relationship between number of people on a person's phone list (xx) and the number of text messages per day (yy), 40 individuals were surveyed and the following information was recorded:
∑i=140xi=700sx=3.5\sum_{i=1}^{40}x_i=700 \quad s_x=3.5
∑i=140yi=3200sy=18.9r=0.67\sum_{i=1}^{40}y_i=3200 \quad s_y=18.9 \quad r=0.67
Regression Line
The equation y^=3+0.025x\hat y=3+0.025x is used to predict the amount of weight lost yy (in kg) by a group of individuals in a study based on the amount of their daily exercise xx (in minutes). Given that 1kg≈2.2lb1kg\approx2.2lb, 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:
xˉ\bar{x}= 60 yˉ\bar{y}= 4.8
sxs_{x}= 38.08 sys_{y}= 2.59 r=0.71r=0.71
Regression Line: Changing Units of $x$
The equation y^=35+1.50x\hat y=35+1.50x 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=0.762sx=278.87sy=60350.59xˉ=1034.84yˉ=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}
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=0.762sx=278.87sy=60350.59xˉ=1034.84yˉ=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}
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=0.762sx=278.87sy=60350.59xˉ=1034.84yˉ=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}
With this linear regression plot
Regression Line: Calculation
One day George wanted to examine the linear relationship between how many hours a student studies (xx) the day before the exam and the exam grade (yy).
He asks 13 random students and collects the following data
r=0.819 sx=3.24 sy=14.24 x‾=8.25  y‾=78.25r=0.819\ \quad s_x=3.24\ \quad s_y=14.24\ \quad\overline{x}=8.25\ \ \overline{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}=35+1.50xx 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 weightLost^\widehat{weight Lost}=3+0.025 (exercise^\widehat{exercise}) 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?
Linear Regression
xˉ\bar{x}= 60 yˉ\bar{y}= 4.8
sxs_{x}= 38.08 sys_{y}= 2.59 r=0.71r=0.71
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.
xˉ\bar{x}= 60 yˉ\bar{y}= 4.8
sxs_{x}= 38.08 sys_{y}= 2.59
Solving for the Regression Line
The equation y^=3+0.025x\hat{y}=3+0.025x is used to predict the amount of weight lost yy, in kilograms, by a group of individuals in a study based on the average amount of daily exercise xx, 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?
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.
More Solving for Correlation Questions:
Correlation Coefficient
Suppose that in a sample of 20 boys and 20 fathers, each father is 15 cm taller than his son. Which of the following is most likely the value of r?
Correlation: Changing Units of X Practice Question
Problem
A researcher correctly calculated the correlation between the shoe size of his subjects, in inches, and the weight of his subjects, in pounds. It turned out to be 0.69. He then decides to change the shoe size units into centimetres. Which of the following is the new value of rr :
Correlation Coefficient
Suppose that in a sample of 20 boys and 20 fathers, each father is 15 cm taller than his son. Which of the following is most likely the value of r?
Correlation
Which of the following is most likely the correlation coefficient between the number of text messages you receive and your daily commute time?
Correlation
Which of the following is most likely the correlation coefficient between the number of cousins you have and the mark you get on your midterm?
Correlation
Calculate the correlation coefficient between time in hours and distance in kilometers given the following data:
sx2=64 sy=10 b = 0.3
Correlation
Which of the following is likely to be the value of the correlation coefficient rr between the variables (1) program of study (such as engineering, sciences, social studies) and (b) amount of homework in hours
Correlations
Calculate the correlation coefficient between time in hours and distance in kilometers given the following data:
sx  2=64,  sy=10,  y^=0.3xs_x^{\ \ 2}=64,\ \ s_y=10,\ \ \hat y=0.3x
Correlations
Which of the following is likely to be the value of the correlation coefficient rr between the variables (1) program of study (such as engineering, sciences, social studies) and (b) amount of homework in hours
Correlation Coefficient
Suppose that in a sample of 20 boys and 20 fathers, each father is 15 cm taller than his son. Which of the following is most likely the value of r?
Correlation
Which of the following is likely to be the value of the correlation coefficient rr between the variables (1) program of study (such as engineering, sciences, social studies) and (b) amount of homework in hours
Correlation: Perfect Linear Correlation
Suppose that in a sample of 20 boys and 20 fathers, each father is 15 cm taller than his son. Which of the following is most likely the value of r?
Correlation: Changing Units of X Practice Question
Problem
A researcher correctly calculated the correlation between the shoe size of his subjects, in inches, and the weight of his subjects, in pounds. It turned out to be 0.69. He then decides to change the shoe size units into centimetres. Which of the following is the new value of rr :
Solving for Correlation
The following data is obtained to assess the relationship between height in cm and shoe size.
The coefficient of correlation is?
Correlation
Solve for the correlation coefficient.
∑x=42\sum_{ }^{ }x=42
∑y=235\sum_{ }^{ }y=235
Correlation
Find the correlation coefficient and interpret it.
∑x=30\sum x=30
∑y=25\sum y=25
Correlation
∑X=33,115\sum X=33,115
∑Y=14,263,500\sum Y=14,263,500
∑XY=15,157,905,000\sum XY=15,157,905,000
Correlation
We want to predict a person’s body fat based on their age. Age and percentage body fat were measured in 18 adult males.
∑x=702∑y=461∑xy=19,871∑x2=31,396∑y2=13,449\begin{array}{ll}\sum x=702\\\sum y=461\\\sum xy=19,871\\\sum x^2=31,396\\\sum y^2=13,449\end{array}
Correlation
(ii) Solve for the correlation coefficient.
Correlation
Answer the following questions.
Practice: Correlation
Answer the following questions.
Correlation
Paul wants to find the correlation between iPhone models (i.e. iPhone 3, 4, 5, 6, 7, 8, X, etc.) and battery life. What is wrong here?
Correlation and Residual Plots
Pamela and Tim are trying to be healthy by taking the stairs instead of the elevator. We want to see there is a relationship between the floor you live on and the time it takes to get to your floor from the lobby. Pamela lives on the 8th floor. Tim lives in the penthouse on the 20th floor.
∑x2=258\sum_{ }^{ }x^2=258
∑y2=687\sum_{ }^{ }y^2=687