Question Type 3: Identifying the appropriate variables for a given contextual information Exercises

Person A studies the effect of weekly junk food consumption on average resting heart rate. The dataset includes variables: $x_1$ = number of candy bars eaten per week, $x_2$ = hours of TV watched per week, $x_3$ = cholesterol level (mg/dL), and $y$ = average resting heart rate (bpm). Identify the explanatory and response variables and justify your choice.

Weekly servings were recorded for week numbers [5, 7, 10, 12, 15, 17] with values [3, 4, 5, 6, 2, 4]. The study focuses only on weeks 10–15. Remove data outside this range and compute the average servings for the relevant weeks.

A dataset for the same study records: weekly fast‐food meals, daily water intake (L), weekly junk food servings, and subject age. For analyzing the effect of junk food consumption on resting heart rate, select the two relevant variables and justify why the others should be excluded.

A dataset lists ages [16, 25, 30, 70, 40] and corresponding resting heart rates [65, 70, 68, 75, 72]. Exclude individuals outside the 18–65 age range and compute the mean resting heart rate of the remaining group.

The weekly junk food servings recorded over six weeks are [2, 35, 3, 4, 5, 3]. Identify any outlier by eye, then recalculate the mean weekly servings after removing the outlier.

Two explanatory variables $x_1$ and $x_2$ have correlation coefficients $r_1=0.65$ and $r_2=0.72$ with the response variable $y$. Which variable should you choose for a simple linear regression and why?

For the chosen variable $x_2$ with correlation coefficient $r_2=0.72$, compute the coefficient of determination and interpret its meaning in this context.

Two candidate explanatory variables are weekly junk food servings $x_1$ and daily sugar intake (g) $x_2$. The corresponding average resting heart rates $y$ (bpm) for five subjects are given by:

Subject A: $(x_1,y)=(4,70)$, $(x_2,y)=(30,70)$

B: $(5,75)$, $(40,75)$

C: $(3,65)$, $(20,65)$

D: $(6,78)$, $(50,78)$

E: $(2,60)$, $(10,60)$

Compute the Pearson correlation coefficient for each candidate ($r_1$ for $x_1,y$ and $r_2$ for $x_2,y$) and decide which variable shows the stronger linear association with $y$.

The distribution of weekly junk food servings is positively skewed. Propose a mathematical transformation to approximate normality before modeling and justify your choice.

In the study, daily exercise minutes is recorded alongside weekly junk food servings and resting heart rate. Identify the potential confounding variable and propose a mathematical method to control for it when modeling resting heart rate.

Using the sugar‐intake data $(x_2,y)$ from the previous question, find the least‐squares regression line y=eta_0+eta_1x_2 predicting heart rate $y$ from sugar intake $x_2$.

Daily sugar intake data over 10 days are [10, 12, 11, 13, 14, 12, 11, 200, 13, 14]. Use the z-score criterion (|z|>3) to determine whether the value 200 should be considered an outlier.