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

Provide an appropriate response. -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). Construct a $95 \%$ prediction interval for $y$ , the yield, given $x = 11$ inches, $\hat { \mathrm { y } } = 4.379 \mathrm { x } + 4.267$ and $\mathrm { s } _ { \mathrm { e } } = 3.529$

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 $95 \%$ confidence interval for the State Board Score a student will get if the student's age is 24 .

Construct Confidence Intervals for a Mean Response -Construct a $95 \%$ confidence interval about the mean value of $y$ , given $x = - 3.5 , \hat { y } = 2.097 x - 0.552$ and $s _ { e } = 0.976$ . -5 -3 4 1 -1 -2 0 2 3 -4 -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 $\hat { \mathrm { y } } = - 2.5 \mathrm { x }$ . -1 -2 -3 -4 2 6 7 10

Conduct Inference on the Slope -Test the claim, at the $\alpha = 0.05$ level of significance, that a linear relation exists between the two variables, for the data below, given that $y = - 2.5 x$ .

The difference between the observed and predicted value of the response variable is a_____

Obtain the Correlation Matrix -A correlation matrix shows the linear correlation among _____variables under consideration in a multiple regression model.

Test Individual Regression Coefficients for Significance -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. Test the significance of the individual predictor variables at $\alpha = 0.05$ .

Construct Confidence Intervals for a Mean Response -When constructing a confidence interval about the mean response of $y$ in a linear regression, the $t$ -distribution is used with _____degrees of freedom.

Construct Confidence Intervals for a Mean Response -In an area of Russia, records were kept on the relationship between the rainfall (in inches) and the yield of wheat (bushels per acre). Construct a $95 \%$ confidence interval about the mean value of $y$ , the yield, given $x = 11$ inches, $\mathrm { y } = 4.379 \mathrm { x } + 4.267$ and $\mathrm { se } _ { \mathrm { e } } = 3.529$ . Rainfall (in inches), 10.5 8.8 13.4 12.5 18.8 10.3 7.0 15.6 16.0 Yield (bushels per acre), 50.5 46.2 58.8 59.0 82.4 49.2 31.9 76.0 78.8

Compute the Standard Error of the Estimate -The data below are the number of absences and the final grades of 9 randomly selected students in an engineering class. Find the standard error of estimate, $s _ { \mathrm { e } }$ , given that $\hat { y } = - 2.75 \mathrm { x } + 96.14$ .

Compute the Standard Error of the Estimate -Find the standard error of estimate, $\mathrm { s } _ { \mathrm { e } }$ , for the data below, given that $\hat { y } = - 1.885 \mathrm { x } + 0.758$ . -5 -3 4 1 -1 -2 0 2 3 -4 11 6 -6 -1 3 4 1 -4 -5 8

Build a Regression Model -A researcher is investigating whether exercise, age, and percent body fat could be good predictors of resting pulse rate. She selects a random sample of women and for each woman records their resting pulse rate, the amount they exercise (on a scale of 1 to 10), age, and percent body fat. The results are shown in the table. Resting Pulse Rate Amount of Exercise Age Percent Body Fat 76 6 22 23 63 8 38 19 82 5 62 25 90 3 54 31 86 2 44 27 77 4 41 24 80 5 59 26 75 4 27 26 58 8 35 16 76 5 76 24 65 9 31 20 85 3 66 28 (a) Construct the correlation matrix. Is there any reason to be concerned with collinearity? Is this what you would expect? (b) Find the least squares regression equation $\hat { y } = b _ { 0 } + b _ { 1 } x _ { 1 } + b _ { 2 } x _ { 2 } + b _ { 3 } x _ { 3 }$ , where $x _ { 1 }$ is Exercise, $x _ { 2 }$ is Age, $x _ { 3 }$ is Percent body fat, and $y$ is the response variable "resting pulse rate". (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? Is this what you would expect? If appropriate, remove an explanatory variable and compute the new least squares regression equation. (g) What is the P-value for your final regression equation? What does this imply?

Conduct Inference on the Slope -If a hypothesis test of the linear relation between the explanatory and the response variable is of the type where $\mathrm { H } _ { 0 } : \beta _ { 1 } = 0 , \mathrm { H } _ { 1 } : \beta _ { 1 } > 0$ , then we are testing the claim that

Provide an appropriate response. -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 $95 \%$ prediction interval about the value of $y$ when $x = 300$ days. Horse Gestation period Life Length x (days) Horse Gestation period Life Length 1 416 24 5 356 22 2 279 25.5 6 403 23.5 3 298 20 7 265 21 4 307 21.5

Compute the Standard Error of the Estimate -The data below are the ages and systolic blood pressures (measured in millimeters of mercury) of 9 randomly selected adults. Find the standard error of estimate, $\mathrm { s } _ { \mathrm { e } }$ , given that $\hat { \mathrm { y } } = 1.488 \mathrm { x } + 60.46$ .

Perform an F-test for Lack of Fit -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. Test the null hypothesis that all coefficients are zero at the $95 \%$ confidence level.

Interpret the Coefficients of a Multiple Regression Equation -Why is it important for the explanatory variables to have a low correlation?

Conduct Inference on the Slope -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. Test the claim, at the $\alpha = 0.05$ level of significance, that a linear relation exists between the two variables, given that $\mathrm { y } = 0.449 \mathrm { x } - 30.27$ .

Compute the Standard Error of the Estimate -In a study of feeding behavior, zoologists recorded the number of grunts of a warthog feeding by a lake in the 15 minute period following the addition of food. The data showing the number of grunts and the age of the warthog (in days) are listed below. Compute the standard error, the point estimate for o. Number of Grunts, y Age (days), x 97 125 75 141 46 155 51 160 70 167 47 174 69 183 24 189 29 195