SCENARIO 13-11
a Weight-Loss Clinic Wants to Use Regression Analysis $$Y = \beta _ { 0 } + \beta _ { 1 } X _ { 1 } + \beta _ { 2 } X _ { 2 } + \beta _ { 3 } X _ { 1 } X _ { 2 } + \varepsilon$$

question 36

Short Answer

SCENARIO 13-11
A weight-loss clinic wants to use regression analysis to build a model for weight loss of a client (measured in pounds).Two variables thought to affect weight loss are client's length of time on the weight-loss program and time of session.These variables are described below:
Y = Weight loss (in pounds)
X1 = Length of time in weight-loss program (in months)
X2 = 1 if morning session, 0 if not
Data for 25 clients on a weight-loss program at the clinic were collected and used to fit the interaction
model: $$Y = \beta _ { 0 } + \beta _ { 1 } X _ { 1 } + \beta _ { 2 } X _ { 2 } + \beta _ { 3 } X _ { 1 } X _ { 2 } + \varepsilon$$ Output from Microsoft Excel follows: $$\begin{array}{l}\begin{array} { l r } \hline \text { Multiple R } & 0.7308 \\\text { R Square } & 0.5341 \\\text { Adjusted R Square } & 0.4675 \\\text { Standard Error } & 43.3275 \\\text { Observations } & 25\\\hline\end{array}\\\text { ANOVA }\\\begin{array} { l r r l l r } \hline & d f & & { \text { SS } } & { \text { MS } } & \text { Significance } F \\\hline \text { Regression } & 3 & 45194.0661 & 15064.6887 & 8.0248 & 0.0009 \\\text { Residual } & 21 & 39422.6542 & 1877.2692 & & \\\text { Total } & 24 & 84616.7203 & &\\\hline\end{array}\\\\\begin{array} { l r r r r r r } \hline & \text { Coefficients } & \text { Standard Error } & { t \text { Stat } } & \text { P-value } & \text { Lower 99\% } & \text { Upper 99\% } \\\hline\text { Intercept } & -20.7298 & 22.3710 & -0.9266 & 0.3646 & -84.0702 & 42.6106 \\\text { Length } & 7.2472 & 1.4992 & 4.8340 & 0.0001 & 3.0024 & 11.4919 \\\text { Morn } & 90.1981 & 40.2336 & 2.2419 & 0.0359 & -23.7176 & 204.1138 \\\text { Length x Morn } & -5.1024 & 3.3511 & -1.5226 & 0.1428 & -14.5905&4.3857\\\hline\end{array}\end{array}$$
-Referring to SCENARIO 13-11, what null hypothesis would you test to determine whether the slope of the linear relationship between weight loss (Y) and time on the program (X1) varies according to time of session? a) $H _ { 0 } : \beta _ { 1 } = 0$
b) $H _ { 0 } : \beta _ { 2 } = 0$
c) $H _ { 0 } : \beta _ { 3 } = 0$
d) $H _ { 0 } : \beta _ { 1 } = \beta _ { 2 } = 0$

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Definitions:

Sedentary Activity

Activities that involve little to no physical movement, resulting in minimal energy expenditure.

Regular Exercise

Engaging in physical activities on a consistent basis to improve or maintain one's physical fitness and overall health.

High Levels

A term that indicates a greater than average quantity, intensity, or degree in a given context.

Team Sports

Sports that involve players working together towards a common objective, typically involving competitive play against another team.