Exam 7: Linear Regression

Consider the four points (20,20),(30,50),(40,30),and (50,60).The least squares line is $\hat{y} = 5 + 50 x$ Explain what "least squares" means using these data as a specific example.

One of the important factors determining a car's fuel efficiency is its weight.This relationship is examined for 11 cars,and the association is shown in the scatterplot below. If a linear model is considered,the regression analysis is as follows: Dependent variable: MPG R-squared = 84.7% VARIABLE COEFFICIENT Intercept 47.1181 Weight -7.34614 The residuals plot is: Based upon the residuals plot,do you think that this linear model is appropriate?

=+ 30 ? 6 9 6 =-84+18

The relationship between the selling price (in dollars)of used Ford Escorts and their age (in years)is analyzed.A regression analysis to predict the price from the age gives the model $\hat{\text { price }} = 14,210 - 1348$ age.You want to sell a 17-year-old Escort.Use the model to determine an appropriate price.Explain any problems.

A random sample of records of electricity usage of homes in the month of July gives the amount of electricity used and size (in square feet)of 135 homes.A regression was done to predict the amount of electricity used (in kilowatt-hours)from size.The residuals plot indicated that a linear model is appropriate.The model is $\hat{\text { usage }} = 1218 + 0.3$ size.What would a negative residual mean for people living in a house that is 2495 square feet?

Managers rate employees according to job performance and attitude.The results for several randomly selected employees are given below. Attitude Performance 59 63 65 69 58 77 76 69 70 64 72 67 78 82 75 87 92 83 87 78

A random sample of records of electricity usage of homes gives the amount of electricity used and size (in square feet)of 135 homes.A regression to predict the amount of electricity used (in kilowatt-hours)from size has an R-squared of 71.3%.The residuals plot indicated that a linear model is appropriate.Write a sentence summarizing what $\mathrm { R } ^ { 2 }$ says about this regression.

A golf ball is dropped from 15 different heights (in cm)and the height of the bounce is recorded (in cm.)The regression analysis gives the model $\hat{\text { bounce }} = 0.4 + 0.72$ drop.A golf ball dropped from 64 cm bounced 1 cm less than expected.How high did it bounce?

Ten students in a graduate program at Carleton University were randomly selected.Their grade point averages (GPAs)when they entered the program were between 11.5 and 12.0.The following data were obtained regarding their GPAs on entering the program versus their current GPAs. Entering GPA (E) Current GPA (C) 11.5 11.6 11.8 11.7 11.6 11.9 11.6 11.6 11.5 11.9 11.9 11.8 12.0 11.7 11.9 11.9 11.5 11.8 11.7 12.0

One of the important factors determining a car's fuel efficiency is its weight.This relationship is examined for 11 cars,and the association is shown in the scatterplot below. If a linear model is considered,the regression analysis is as follows: Dependent variable: MPG R-squared = 84.7% VARIABLE COEFFICIENT Intercept 47.1181 Weight -7.34614 What does the slope say about this relationship?

A golf ball is dropped from 15 different heights (in cm)and the height of the bounce is recorded (in cm.)The regression analysis gives the model $\hat{\text { bounce }}$ = 0.3 + 0.71 drop.Interpret the meaning of the y-intercept.

A random sample of records of electricity usage of homes gives the amount of electricity used in July and size (in square feet)of 135 homes.A regression was done to predict the amount of electricity used (in kilowatt-hours)from size.Suppose the linear model is appropriate.The model is $\hat{\text { usage }} = 1248 + 0.6$ size.Explain what the slope of the line says about the electricity usage and home size.

A psychologist does an experiment to determine whether an outgoing person can be identified by his or her handwriting.She claims that the $\mathrm { R } ^ { 2 }$ of 89% shows that this linear model is appropriate.

The relationship between the number of games won by an NHL team and the average attendance at their home games is analyzed.A regression analysis to predict the average attendance from the number of games won gives the model $\hat{\text { attendance }} = - 2100 + 193$ wins.Predict the average attendance of a team with 58 wins.

A random sample of records of electricity usage of homes gives the amount of electricity used in July and size (in square feet)of 135 homes.A regression was done to predict the amount of electricity used (in kilowatt-hours)from size.Suppose the linear model is appropriate.What units does the slope have?

Using advertised prices for used Ford Escorts a linear model for the relationship between a car's age and its price is found.The regression has an $\mathrm { R } ^ { 2 }$ = 87.7%.Write a sentence summarizing what $\mathrm { R } ^ { 2 }$ says about this regression.

List all the regression assumptions and conditions described in Chapter 7 of your textbook.

A random sample of records of electricity usage of homes in the month of July gives the amount of electricity used and size (in square feet)of 135 homes.A regression was done to predict the amount of electricity used (in kilowatt-hours)from size.The residuals plot indicated that a linear model is appropriate.The model is $\hat{\text { usage }} = 1287 + 0.3$ size.The people in a house that is 2347 square feet used 500 kilowatt-hours less than expected.How much did they use?

A forester would like to know how big a maple tree might be at age 50 years.She gathers data from some trees that have been cut down,and plots the diameters (in inches)of the trees against their ages (in years).First she makes a linear model.The scatterplot and residuals plot are shown.If she uses this model to try to predict the diameter of a 50-year old maple tree,would you expect that estimate to be fairly accurate,too low,or too high? Explain.