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

Compute the Standard Error of the Estimate -Find the standard error of estimate, $\mathrm { s } _ { \mathrm { e } }$ , for the data below, given that $\hat { \mathrm { y } } = 2.097 \mathrm { x } - 0.552$ . -5 -3 4 1 -1 -2 0 2 3 -4 -10 -8 9 1 -2 -6 -1 3 6 -8

Construct Confidence and Prediction Intervals -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 data for this class is given below. Construct a $95 \%$ prediction interval for the Rope Climb Time if the student's Pull-ups are $22 .$

Construct Confidence and Prediction Intervals -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. Construct a $95 \%$ prediction interval for the State Board Score a student will get if the student's age is 24 .

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, $x$ , the number of hours viewing television, $x 2$ , and the number of hours on their cell phone, $x _ { 3 }$ , per day as $\hat { \mathrm { y } } = 125.5 + 3.26 \mathrm { x } _ { 1 } + 4.27 \mathrm { x } _ { 2 } + 1.75 \mathrm { x } _ { 3 }$ . What is a student's expected weight change if the amount of television is increased by 1 hour, the number of hours watching television and talking on their cell phone stayed the same?

Build a Regression Model -Can happiness be predicted? A researcher is investigating whether job satisfaction, income, and health could be good predictors of happiness. She selects a random sample of working adults and obtains the following information for each person. Each person is asked to rate their happiness and their job satisfaction on a scale of 1 to 10 . Each person is asked their annual income. Finally, each person's overall physical health is evaluated by a doctor and rated on a scale of 1 to 20 . The results are shown in the table. Happiness Job satisfaction Annual Income (thousands of dollars) Health 8 6 38 14 6 5 47 17 4 2 19 12 7 5 22 11 2 3 36 5 9 7 44 14 4 3 75 8 6 8 24 13 7 4 52 14 3 3 27 5 5 4 105 12 8 9 44 7 (a) Construct the correlation matrix. Is there any reason to be concerned with collinearity? (b) Find the least squares regression equation $\mathrm { y } = \mathrm { b } _ { 0 } + \mathrm { b } _ { 1 } \mathrm { x } _ { 1 } + \mathrm { b } _ { 2 } \mathrm { x } _ { 2 } + \mathrm { b } _ { 3 } \mathrm { x } _ { 3 }$ , where $\mathrm { x } _ { 1 }$ is job satisfaction, $\mathrm { x } _ { 2 }$ is income, $x _ { 3 }$ is health, and $y$ is the response variable "happiness". (c) Test $\mathrm { H } _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = 0$ versus $\mathrm { H } _ { 1 }$ : at least one of the $\beta _ { \mathrm { i } } \neq 0$ at the $\alpha = 0.05$ level of significance. (d) Test the hypotheses $\mathrm { H } _ { 0 } : \beta _ { 1 } = 0$ versus $\mathrm { H } _ { 1 } : \beta _ { 1 } \neq 0 , \mathrm { H } _ { 0 } : \beta _ { 2 } = 0$ versus $\mathrm { H } _ { 1 } : \beta _ { 2 } \neq 0$ , and $\mathrm { H } _ { 0 } : \beta _ { 3 } = 0$ versus $\mathrm { H } _ { 1 } : \beta _ { 3 } \neq 0$ at the $\alpha = 0.05$ level of significance. Should any of the explanatory variables be removed from the model? If so, which one? Why? (e) Determine the least squares regression equation with the explanatory variable identified in part (d) removed. (f) Are both slope coefficients significantly different from zero? If not, remove the appropriate explanatory variable and compute the new least squares regression equation. (g) How does the $\mathrm { P }$ -value for your final regression equation compare with the $\mathrm { P }$ -value for the original equation in part (b)? What does this imply?

Perform an F-test for Lack of Fit -By saying that we reject the null hypothesis in a hypothesis test of the coefficients of a multiple linear regression we are implying that

Compute the Standard Error of the Estimate -The data below are the final exam scores of 10 randomly selected engineering students and the number of hours they slept the night before the exam. Find the standard error of estimate, se given that $\mathrm { y } = 5.044 \mathrm { x } + 56.11$ . Hours, 3 5 2 8 2 4 4 5 6 3 Scores, 65 80 60 88 66 78 85 90 90 71

Perform an F-test for Lack of Fit -A minor league baseball team posted the following data for its 16 best players. Test the null hypothesis that all coefficients are zero at the $95 \%$ confidence level.

Determine R^2 and Adjusted R^2 -A minor league baseball team posted the following data for its 16 best players. 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 At Bats, $x _ { 2 }$ is Strikeouts, and $y ^ { 2 }$ is the response variable "Hits".

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 $\mathrm { R } ^ { 2 }$ for the multiple regression model $\hat { \mathrm { y } } = \mathrm { b } _ { 0 } + \mathrm { b } _ { 1 } \mathrm { x } _ { 1 } + \mathrm { b } _ { 2 } \mathrm { x } _ { 2 }$ where $\mathrm { x } _ { 1 }$ is Push Ups, $\mathrm { x } _ { 2 }$ is Pull Ups, and $\mathrm { y }$ is the response variable "Rope Climb".

Compute the Standard Error of the Estimate -In an area of the Great Plains, records were kept on the relationship between the rainfall (in inches) and the yield of wheat (bushels per acre). Find the standard error of estimate, se given that $\mathrm { y } = 4.379 \mathrm { x } + 4.267$ .

Construct Confidence Intervals for a Mean Response -In order for a company's employees to work at a foreign office, they must take a test in the language of the country where they plan to work. The data below shows the relationship between the number of years that employees have studied a particular language and the grades they received on the proficiency exam. Construct a $95 \%$ confidence interval about the mean value of $\mathrm { y }$ , given $\mathrm { x } = 2.5 , \mathrm { y } = 6.91 \mathrm { x } + 46.26$ , and $\mathrm { s } e = 4.578$ . Number of years, 3 4 4 5 3 6 2 7 3 Grades on test, 61 68 75 82 73 90 58 93 72

Construct a Confidence Interval about the Slope of the Least -squares Regression Model -Construct a $99 \%$ confidence interval about the slope of the true least-squares regression line, for the data below, given that $\hat { \mathrm { y } } = 2.097 \mathrm { x } - 0.552$ . x -5 -3 4 1 -1 -2 0 2 3 -4 y -10 -8 9 1 -2 -6 -1 3 6 -8

Compute the Standard Error of the Estimate -Find the standard error of estimate, $\mathrm { s } _ { \mathrm { e } }$ , for the data below, given that $\hat { y } = - 0.206 \mathrm { x } + 2.097$ . -5 -3 4 1 -1 -2 0 2 3 -4 11 -6 8 -3 -2 1 5 -5 6 7

Construct Confidence Intervals for a Mean Response -The data below are the scores of 10 randomly selected students from a statistics class and the number of hours they slept the night before the exam. Construct a $95 \%$ confidence interval about the mean value of $y$ , the score on the final exam, given $x = 7$ hours, $\mathrm { y } = 5.044 \mathrm { x } + 56.11$ and $\mathrm { se } = 6.305$ . Hours, 3 5 2 8 2 4 4 5 6 3 Scores, 65 80 60 88 66 78 85 90 90 71

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

Conduct Inference on the Slope -The data below are the ages and systolic blood pressures (measured in millimeters of mercury) of 9 randomly selected adults. Test the claim, at the $\alpha = 0.01$ level of significance, that a linear relation exists between the two variables, given that $\mathrm { y } = 1.488 \mathrm { x } + 60.46$

Compute the Standard Error of the Estimate -Find the standard error of estimate, $s _ { e }$ , for the data below, given that $\hat { y } = 2 x + 1$ .

Determine R^2 and Adjusted R^2 -Twelve nursing students are set to graduate and the registration clerk at the nursing school wonders if a student's age and GPA would be good predictors of their state board score. She obtains the following data. 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 Age, $x _ { 2 }$ is GPA, and $y$ is the response variable "state board score".

Obtain the Correlation Matrix -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 data for this class is given below. What is the correlation between pull-ups and the time to climb the rope?