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Correlation and Residual Plots

Related Topics

Wize University Statistics Textbook > Simple Linear Regression

Predictions and Residual Plots

8 Activities

Wize University Statistics Textbook > Inference for Linear Regression

Simple Linear Regression Analysis

3 Activities

Wize University Statistics Textbook > Scatterplots, Association, and Correlation

Coefficient of Determination

3 Activities

Wize University Statistics Textbook > Scatterplots, Association, and Correlation

Solving for Correlation

5 Activities

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

Which of the following 3 statements below are true? (Select 3 correct statements.)



More Predictions and Residual Plots Questions:
You wish to predict the number of times someone logs in their email a day. The estimated linear regression model is Logins = 5 + 1.3(work)Logins\ =\ 5\ +\ 1.3\left(work\right)where Logins is the number of times a person logs in their email a week and work is the number of hours of work done a week. Logan works 30 hours a week and checks his email 49 times a week. What is his residual?
Predictions and Residual Plots
You wish to predict the number of times someone logs in their email a day.
The estimated linear regression model is Logins=5+1.3(work)Logins=5+1.3\left(work\right)
where Logins is the number of times a person logs in their email a week and
Predictions and Residual Plots
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
Predictions and Residual Plots
Suppose the following summary data was given:
x-variable: μx=10, sx=2\mu_x=10,\ s_x=2
y-variable: μy=250, sy=10\mu_y=250,\ s_y=10
Linear Regression: Residual Plots
The equation y^=35+1.50x\hat{y}=35+1.50x is used to predict the hourly wage yy of someone with xx years of experience.
Elizabeth has 4 years of experience and earns $36 per hour. What is the residual?
Predictions: Residual Plots
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
Predictions and Residual Plots
In linear regression, which of the following indicates that we should fit the scatter plot with a line of best fit?
Predictions and Residual Plots
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 16 units:
(a) Estimate the linear regression model.
(b) Unit #3 is 1250 sqft, what is his predicted selling price? Compute the residual for this estimate.
Predictions and Residual Plots
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 16 units, here are the results and the regression output:
(a) Estimate the linear regression model.
(b) Unit #3 is 1250 sqft, what is his predicted selling price? Compute the residual for this estimate.
Predictions and Residual Plots
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
Predictions and Residual Plots
Cindy calculated residuals and find that all of them are negative. Which of the following is true?
Predictions and residual plots
The following assertions about the three residual are True or False
Residual Plot A: No problem – unbiased and homoscedastic (constant variance)
Residual Plot B: Heteroscedasticity – we see a “fan”, indication of non-constant variance across the range of values
Predictions: Residual Plots
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=700   sx=3.5\sum_{i=1}^{40}x_i=700\quad\ \ \ s_x=3.5
∑i=140yi=3200  sy=18.9  r=0.67\sum_{i=1}^{40}y_i=3200\ \ \quad s_y=18.9\quad\ \ r=0.67
Residual Plots: Calculation
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
Predictions and Residual Plots
You wish to predict the number of times someone logs in their email a day. The estimated linear regression model is Logins = 5 + 1.3(work)Logins\ =\ 5\ +\ 1.3\left(work\right)where Logins is the number of times a person logs in their email a week and work is the number of hours of work done a week. Logan works 30 hours a week and checks his email 49 times a week. What is his residual?
Predictions and Residual Plots
The regression model y=4000−2.7xy=4000-2.7x is used to estimate the annual number of cars going over the speed limit (y) based on the number of police officers on the road. Estimate the number of cars going over the speed limit when there are 40 police officers on the road.
Predictions and Residual Plots: Regression Diagnostics Question
Let’s take a look at our residual plot in our example:
Regression Diagnostics (check all that applies):
[ ] Outliers
Predictions: Residual Plots
(v) Mike actually has 8% body fat. What is the residual?
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.
Residual Plots
Here are three residual plots. Which of the following is true?
Predictions and Residual Plots
The equation y^\hat{y}=35+1.50xx is used to predict the hourly wage yy of someone with xx years of experience.
(c) Elizabeth has 4 years of experience and earns $36. What is the residual?
Simple Linear Regression
Here is a residual plot for selling price (Y) vs. size of a condo in square-feet (X).
Which condition is violated?
More Simple Linear Regression Analysis Questions:
Linear Regression
Parts (a) to (c) are based on the following information.
The equation y^\hat{y}=35+1.50xx is used to predict the hourly wage yy of someone with xx years of experience. The correlation is r=0.56r=0.56. Suppose that xx is changed to months of experience.
(a) Which of the following will be the correct regression line?
Linear Regression
Parts (a) to (c) are based on the following information.
The equation y^\hat{y}=35+1.50xx is used to predict the hourly wage yy of someone with xx years of experience. The correlation is r=0.56r=0.56. Suppose that xx is changed to months of experience.
(a) Which of the following will be the correct regression line?
Linear Regression
Parts (a) to (c) are based on the following information.
The equation y^\hat{y}=35+1.50xx is used to predict the hourly wage yy of someone with xx years of experience. The correlation is r=0.56r=0.56. Suppose that xx is changed to months of experience.
(a) Which of the following will be the correct regression line?
Linear Regression
Parts (a) to (c) are based on the following information.
The equation y^\hat{y}=35+1.50xx is used to predict the hourly wage yy of someone with xx years of experience. The correlation is r=0.56r=0.56. Suppose that xx is changed to months of experience.
(a) Which of the following will be the correct regression line?
Linear Regression
Parts (a) to (c) are based on the following information.
The equation y^\hat{y}=35+1.50xx is used to predict the hourly wage yy of someone with xx years of experience. The correlation is r=0.56r=0.56. Suppose that xx is changed to months of experience.
(a) Which of the following will be the correct regression line?
Linear Regression
Parts (a) to (c) are based on the following information.
The equation y^\hat{y}=35+1.50xx is used to predict the hourly wage yy of someone with xx years of experience. The correlation is r=0.56r=0.56. Suppose that xx is changed to months of experience.
(a) Which of the following will be the correct regression line?
Linear Regression
Parts (a) to (c) are based on the following information.
The equation y^\hat{y}=35+1.50xx is used to predict the hourly wage yy of someone with xx years of experience. The correlation is r=0.56r=0.56. Suppose that xx is changed to months of experience.
(a) Which of the following will be the correct regression line?
Linear Regression Analysis
The regression model y=4000−2.7xy=4000-2.7x is used to estimate the annual number of
cars going over the speed limit (y) based on the number of police officers on
the road. Estimate the number of cars going over the speed limit when there are
Linear Regression: Residual Plots
The equation y^=35+1.50x\hat{y}=35+1.50x is used to predict the hourly wage yy of someone with xx years of experience.
Elizabeth has 4 years of experience and earns $36 per hour. What is the residual?
Linear regression
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}
Linear regression
Savage question!
(iv) What body fat % would you predict for Mike who is turning 32 years old?
∑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}
Linear Regression Analysis
Suppose we want to see if there is a relationship between the size of a condo unit in Twin Pines and its selling price. We randomly sampled 32 units:
What is the explanatory variable (X)?
Simple Linear Regression Analysis
Given the information below, find the slope and intercept of the regression line. [Provide at least one decimal place, e.g. 8.2. Include negative sign if negative, e.g. -3.9.]
Linear Regression
Parts (a) to (c) are based on the following information.
The equation y^\hat{y}=35+1.50xx is used to predict the hourly wage yy of someone with xx years of experience. The correlation is r=0.56r=0.56. Suppose that xx is changed to months of experience.
(a) Which of the following will be the correct regression line?
Simple Linear Regression: Correlation
True or false? Enter T if True, F if False. (Use capital letters: T, F)
1. If 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.
T
Predictions and Residual Plots
The equation y^\hat{y}=35+1.50xx is used to predict the hourly wage yy of someone with xx years of experience.
(c) Elizabeth has 4 years of experience and earns $36. What is the residual?
Simple Linear Regression
Here is a residual plot for selling price (Y) vs. size of a condo in square-feet (X).
Which condition is violated?
More Coefficient of Determination Questions:
Coefficient of Determination
In a military aircraft the correlation between altitude (height) in feet (xx) and the number of hits (yy) on a certain target is 0.59. What is the correct conclusion?
Coefficient of Determination
In a military aircraft the correlation between altitude (height) in feet (xx) and the number of hits (yy) on a certain target is 0.59. What is the correct conclusion?
Coefficient of Determination
The following data is collected about 2 quantitative variables.
xˉ\bar{x}= 6 yˉ\bar{y}= 7
sxs_{x}= 2 sys_{y}= 2.6
Coefficient of Determination
(iii) What percent in the variation of body fat could be explained by age?
Coefficient of Determination
We want to see if the amount of training a employee receives is associated with the number of mistakes they make on the job. The coefficient of determination is 0.49.
True or false? Enter T for True; F for False.
Coefficient of determination
If the coefficient of determination r2r^2 is very strong, then xx and yy have a linear relationship.
Correlation: Coefficient of Determination
True or false?
About 81% of the variation of quantities sold can be explained by price. The correlation between price and sales revenue is r=0.9r=0.9.
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 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.
∑(xi−x‾)(yi−y‾)=264.6\sum_{ }^{ }\left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)=264.6
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?