Build a Regression Model
-A Researcher Is Investigating Whether Exercise

question 33

Short Answer

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.
$\begin{array}{c|c|c|c}\text { Resting Pulse Rate } & \text { Amount of Exercise } & \text { Age } & \text { Percent Body Fat } \\\hline 76 & 6 & 22 & 23 \\63 & 8 & 38 & 19 \\82 & 5 & 62 & 25 \\90 & 3 & 54 & 31 \\86 & 2 & 44 & 27 \\77 & 4 & 41 & 24 \\80 & 5 & 59 & 26 \\75 & 4 & 27 & 26 \\58 & 8 & 35 & 16 \\76 & 5 & 76 & 24 \\65 & 9 & 31 & 20 \\85 & 3 & 66 & 28\end{array}$

(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?

---

Definitions:

Culture

The sum of attitudes, customs, and beliefs that distinguishes one group of people from another, transmitted through languages, material objects, ritual, institutions, and art, from one generation to the next.

Social Rules

Agreed-upon guidelines that dictate the behavior expected of individuals within a society or group.

Conventions

Established norms, rules, or practices that dictate certain behaviors or actions within a society or group.

Difference-equals-deficit

A biased belief that differences in cultural or individual characteristics are inherently inferior.