Exam 14: Inference of the Least-Squares Regression Model and Multiple Regression

Determine R^2 and Adjusted R^2 -Twelve nursing students are set to graduate and the registration clerk at the nursing school wonders whether a student's age and GPA would be good predictors of their state board score. She obtains the following data. Determine $\mathrm { R } ^ { 2 }$ for the multiple regression model $\hat { y } = b _ { 0 } + b _ { 1 } x _ { 1 } + b _ { 2 } x _ { 2 }$ where $x _ { 1 }$ is Age, $x _ { 2 }$ is $G P A _ { \text {, and } }$ is the response variable "state board score".

Interpret the Coefficients of a Multiple Regression Equation -Twelve nursing students are set to graduate and the registration clerk at the nursing school wonders if there is a correlation between a student's age, their GPA and their state board score. She collects data about the 12 students and determines the regression equation to be State Board $= 255 + 12.7$ Age $+ 4.6$ GPA. What would she expect a 27 year-old student with a GPA of $2.6$ to achieve on the state board examination? Round your answer to the nearest whole number.

Test Individual Regression Coefficients for Significance -A minor league baseball team posted the following data for its 16 best players. Test the significance of the individual predictor variables at $\alpha = 0.05$ .

Obtain the Correlation Matrix -A minor league baseball team posted the following data for its 16 best players. What is the correlation between the number of strikeouts and the number of hits?

Construct a Confidence Interval about the Slope of the Least -squares Regression Model -Construct a $90 \%$ confidence interval about the slope of the true least-squares regression line, for the data below, for the data below, given that $\hat { y } = - 1.885 x + 0.758$ . -5 -3 4 1 -1 -2 0 2 3 -4 11 6 -6 -1 3 4 1 -4 -5 8

Provide an appropriate response. -Construct a 95\% prediction interval for $y$ given $x = - 3.5 , y = 2.097 x - 0.552$ and $\mathrm { s_e } { = 0.976 \text {. } }$ -5 -3 4 1 -1 -2 0 2 3 -4 -10 -8 9 1 -2 -6 -1 3 6 -8

Interpret the Coefficients of a Multiple Regression Equation -Given the regression equation that relates a student's weight to the number of hours spent on the computer, $x _ { 1 }$ , the number of hours viewing television, $x _ { 2 }$ , and the number of hours on their cell phone, $x _ { 3 }$ , as $\hat { y } = 125.5 + 3.26 x _ { 1 } + 4.27 x _ { 2 } + 1.75 x _ { 3 }$ . What is a student's weight if the student spends 4 hours on the computer, 2 hours viewing television, and 3 hours on their cell phone?

Construct Confidence and Prediction Intervals -A minor league baseball team posted the following data for its 16 best players. Construct a $95 \%$ confidence interval for the number of hits player will get if the player is at bat 53 times.

Use Technology to Find a Multiple Regression Equation -Twelve nursing students are set to graduate and the registration clerk at the nursing school wonders if there is a correlation between a student's age, their GPA and their state board score. She collects data about the 12 students. The data is given below. What is the regression equation?

Construct a Confidence Interval about the Slope of the Least -squares Regression Model -A breeder of Thoroughbred horses wishes to model the relationship between the gestation period and the length of life of a horse. The breeder believes that the two variables may follow a linear trend. The information in the table was supplied to the breeder from various thoroughbred stables across the state. Construct a $90 \%$ confidence interval about the slope of the true least-squares regression line. Horse Gestation Life Horse Gestation Life period x (days) Length y (years) period Length 1 416 24 5 x (days) y (years) 2 279 25.5 6 403 22 3 298 20 7 265 21 4 307 21.5

Construct Confidence Intervals for a Mean Response -The data below are the temperatures on randomly chosen days during the summer and the number of employee absences at a local company on those days. Construct a $95 \%$ confidence interval about the mean value of $\mathrm { y }$ , the number of days absent, given $\mathrm { x } = 95$ degrees, $\mathrm { y } = 0.449 \mathrm { x } - 30.27$ and $\mathrm { se } = 0.934$ .

Construct a Confidence Interval about the Slope of the Least -squares Regression Model -Construct a $95 \%$ confidence interval about the slope of the true least-squares regression line, for the data below, given that $\hat { y } = - 2.5 x$ . -1 -2 -3 -4 2 6 7 10

Conduct Inference on the Slope -Test the claim, at the $\alpha = 0.10$ level of significance, that a linear relation exists between the two variables, for the data below, given that $\mathrm { y } = - 1.885 \mathrm { x } + 0.758$ . -5 -3 4 1 -1 -2 0 2 3 -4 11 6 -6 -1 3 4 1 -4 -5 8

Interpret the Coefficients of a Multiple Regression Equation -A gym teacher uses three exercises to increase arm strength: push-ups, pull-ups, and rope climbing. He has 10 students in small class. The exercises consist of doing as many push-ups as possible in 2 minutes. After a rest the student does as many push-ups as possible in 2 minutes. After a rest the student climbs a rope and the time is recorded in seconds. The regression equation for this class is Rope climb $= 117$ - 1.91 Push-up - 1.06 Pull-ups A new student enters the class and is able to do 20 push-ups and 15 pull-ups, what is this student's expected time to climb the rope? Round your answer to the next whole second.

Compute the Standard Error of the Estimate -A manager wishes to determine the relationship between the number of years her sales representatives have been with the firm and their average monthly sales (in thousands of dollars). Find the standard error of estimate, $\mathrm { s } _ { \mathrm { e } }$ , given that $\mathrm { y } = 3.53 \mathrm { x } + 37.92$ .

The least-squares regression model for one explanatory variable is given by the equation

Determine R^2 and Adjusted R^2 -A gym teacher uses three exercises to increase arm strength: push-ups, pull-ups, and rope climbing. He has 10 students in his class. He records the number of push ups each student is able to do in 2 minutes, the number of pull ups each student is able to do in 2 minutes, and the time (in seconds) for each student to climb a rope. The data are shown in the table. The teacher wonders whether the number of push ups and the number of pull ups would be good predictors of the time to climb the rope. Determine the adjusted $R ^ { 2 }$ for the multiple regression model $\hat { y } = b _ { 0 } + b _ { 1 } x _ { 1 } + b _ { 2 } x _ { 2 }$ where $x _ { 1 }$ is Push Ups, $x _ { 2 }$ is Pull Ups, and $y$ is the response variable "Rope Climb".

Construct a Confidence Interval about the Slope of the Least -squares Regression Model -The data below are the temperatures on randomly chosen days during the summer and the number of employee absences at a local company on those days. Construct a $95 \%$ confidence interval about the slope of the true least-squares regression line, for the data below, given that $\mathrm { y } = 0.449 \mathrm { x } - 30.27$ .

Perform an F-test for Lack of Fit -Twelve nursing students are set to graduate and the registration clerk at the nursing school wonders if there is a correlation between a student's age, their GPA and their state board score. She collects data about the 12 students. The data is given below. Test the null hypothesis that all coefficients are zero at the $95 \%$ confidence level.

Provide an appropriate response. -The data below are the temperatures on randomly chosen days during the summer and the number of employee absences at a local company on those days. Construct a $95 \%$ prediction interval for $y$ , the number of days absent, given $x = 95$ degrees, $\hat { y } = 0.449 \mathrm { x } - 30.27$ and $\mathrm { s } _ { \mathrm { e } } = 0.934$ .