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) $y$ = average resting heart rate (bpm)

Identify the explanatory and response variables and justify your choice.

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

A set of daily sugar intake data (in grams) over 10 days is given below:

$[10, 12, 11, 13, 14, 12, 11, 200, 13, 14]$

Use the $z$-score criterion ($|z| > 3$) to determine whether the value $200$ should be considered an outlier. Show all your calculations.

Two explanatory variables $x_1$ and $x_2$ have correlation coefficients $r_1 = 0.65$ and $r_2 = 0.72$ with the response variable $y$.

State which variable should be chosen for a simple linear regression model and justify your choice.

In a study investigating the relationship between weekly junk food servings and resting heart rate, daily exercise minutes are also recorded for each participant.

Identify the potential confounding variable and propose a mathematical method to control for it when modeling resting heart rate.

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

Subject B: $(x_1, y) = (5, 75)$, $(x_2, y) = (40, 75)$

Subject C: $(x_1, y) = (3, 65)$, $(x_2, y) = (20, 65)$

Subject D: $(x_1, y) = (6, 78)$, $(x_2, y) = (50, 78)$

Subject E: $(x_1, y) = (2, 60)$, $(x_2, y) = (10, 60)$

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 $y$.

Using the summary statistics provided, find the least-squares regression line $y = \beta_0 + \beta_1 x_2$ predicting heart rate $y$ from sugar intake $x_2$. Give the coefficients $\beta_0$ and $\beta_1$ to three significant figures.

A dataset lists ages $[16, 25, 30, 70, 40]$ and corresponding resting heart rates $[65, 70, 68, 75, 72]$. By excluding individuals with ages outside the $18\text{--}65$ range, calculate the mean resting heart rate of the remaining group.

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.

The weekly junk food servings recorded over six weeks are $2, 35, 3, 4, 5, 3$. Identify the outlier by eye and calculate the mean weekly servings after removing this outlier.

A dataset for a study records the following variables: weekly fast-food meals, daily water intake ($L$), weekly junk food servings, subject age, and resting heart rate (bpm).

To analyze the effect of junk food consumption on resting heart rate, select the two relevant variables and justify why the others should be excluded.

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