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Predictions and Residual Plots

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Wize University Statistics Textbook > Simple Linear Regression

Predictions and Residual Plots

8 Activities

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Mark Yourself Question
  1. Grab a piece of paper and try this problem yourself.
  2. When you're done, check the "I have answered this question" box below.
  3. View the solution and report whether you got it right or wrong.
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.
(c) Interpret the meaning of the slope and the intercept.
(d) Find the value of the coefficient of determination.
(e) Find the value of the standard deviation of the regression mode, interpret the value.
(f) Using the T-test, determine if we can conclude that a linear relationship exists between condo size and selling price at the 1% significance level.
(g) Using the F-test, determine if we can conclude that a linear relationship exists between condo size and selling price at the 1% significance level.
(h) Construct a 99% confidence interval for the slope.
(i) In a particular area, the mean size of apartments is 1047sqft. With 99% confidence, predict the mean selling price of houses in that area.
(j) Irma, an apartment owner, is trying to sell her 955 sqft condo. With 99% confidence, predict the selling price of her condo.
(k) A residual plot for selling price vs. sqft was created for our sample and 16 additional observations, what can you say about the plot?

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
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.
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
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?