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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

Wize University Statistics Textbook > Scatterplots, Association, and Correlation

Correlation

2 Activities

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
  • correlation coefficient: r=−0.13r=-0.13
If one observation has an x-value of 8 and a y-value of 260, what do we predict the y-value of another observation with an x-value of 12 to be?
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
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.
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 $36. What is the residual?
Simple Linear Regression
Here is a residual plot for selling price (Y) vs. size of a condo in square-feet (X).
Which condition is violated?
More Correlation Questions:
The correlation between the number of hours of operation and sales in dollars at Pino's Pizza is 0.43. Which of the following is true?
Correlation and Regression Line
If r=−1.00r=-1.00, then 100% of the data points fall exactly on the regression line and the slope cannot be equal or greater than 0.
Correlation
The correlation between the number of hours of operation and sales in dollars at Pino's Pizza is 0.43. Which of the following is true?
Correlation
Numbers in a data set are in dollar units ($). Which of the following measures will not be in dollar units?
Correlation
Which of the following statements is/or TURE? (select all that apply)
Correlation
True or False: A correlation coefficient of 0.6 means that 60% of the data points in the scatter plot have a linear relationship.
Correlation
If we double all the YY data and without changing the XX data, the correlation will not change but the slope will get steeper.
Correlation
If R2=0.86R^2=0.86, then there must be a strong, positive correlation between XX and YY.
Correlation
The correlation between the number of hours of operation and sales in dollars at Pino's Pizza is 0.43. Which of the following is true?
Hypothesis Testing: Correlation
For the following statements, select true or false. Enter T for True, F for False.
Correlation
If we swap xx and yy, what will happen to the correlation coefficient rr?
Correlation
The correlation between midterm grades and final exam grades is r=0.88r=0.88. The correlation between assignment grades and final exam grades is r=0.44r=0.44.
Which of the following is false?
Correlation
In a plot, if we swap xx and yy, rr will not change.
Correlation
r>0r>0; large values of xx are associated with large values of yy.
Correlation
r=−0.91r=-0.91; the correlation is very weak.
Correlation
r<0r<0; increasing xx will cause yy to decrease.
Correlation
r=−1.00r=-1.00; the correlation is perfectly negative.
Correlation
r=2.00r=2.00; the correlation is very strong.
Correlation
r=0.33r=0.33; the correlation is positive, but weak.
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.
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
Correlation
The correlation is 0.56 when the unit for xx is years. What will the correlation coefficient be after xx is changed from years to months of experience? [Enter your answer with 2 decimal places, e.g. 0.36]
Correlation
Correlation rr has no units unless xx and yy have the same units.