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Question 1

Using the 15-week cleaned dataset of weekly junk food servings and idle heart rate below, compute the Pearson correlation coefficient $r$ .

Weekly junk food servings ( $x$ ): $[3, 5, 4, 6, 7, 5, 4, 8, 2, 6, 5, 4, 7, 6, 5]$ Idle heart rate ( $y$ ): $[70, 72, 69, 71, 73, 70, 68, 74, 67, 71, 70, 69, 73, 72, 70]$

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Question 2

Test whether the slope $m=0.83$ of the regression line is significantly different from zero at $\alpha = 0.05$ , given $t = \frac{m}{SE_m} = \frac{0.83}{0.15} = 5.53$ with 13 degrees of freedom.

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Question 3

A study was conducted to determine the relationship between the number of weekly junk food servings ( $x$ ) and the idle heart rate ( $y$ , in beats per minute) of 15 individuals. The following summary statistics were obtained:

\begin{aligned} \sum_{i=1}^{15} (x_i - \bar{x})^2 &= 35.74 \\ \sum_{i=1}^{15} (x_i - \bar{x})(y_i - \bar{y}) &= 29.6 \\ \bar{x} &= 5.13 \\ \bar{y} &= 70.8 \end{aligned}

Fit a least-squares regression line $y = mx + c$ predicting idle heart rate from weekly junk food servings. Calculate the values of $m$ and $c$ , and state the regression equation.

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Question 4

From the cleaned dataset of 15 weeks, the weekly junk food servings are 3, 5, 4, 6, 7, 5, 4, 8, 2, 6, 5, 4, 7, 6, 5. Calculate the mean and standard deviation of junk food servings.

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Question 5

Person A has recorded the following variables over 10 weeks:

Weekly junk food servings: $[3, 5, 4, 6, 7, 5, 4, 8, 2, 6]$
Daily screen time (hours): $[2, 3, 2.5, 4, 3.5, 3, 2, 4.5, 2, 3]$
Average resting heart rate (bpm): $[70, 72, 69, 71, 73, 70, 68, 74, 67, 71]$

Which of these variables should Person A include to investigate the effect of weekly junk food consumption on average resting heart rate, and why?

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Question 6

Person A also recorded weekly exercise hours. Explain how weekly exercise could act as a confounding variable when analyzing the relationship between junk food consumption and idle heart rate, and suggest a mathematical method to adjust for it.

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Question 7

Over 12 weeks, Person A’s weekly junk food servings are recorded as follows: $\{4, 5, 6, 100, 5, 6, 7, 5, 4, 6, 5, 120\}$ Identify and remove any outliers in this data set using the interquartile range (IQR) method.

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Question 8

Given 18 weeks of data labeled week 1 through 18, Person A wants only the most recent 12 weeks. Write a mathematical expression to represent this subset for analysis.

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Question 9

After selecting the last 12 weeks, Person A notices one value of weekly junk food servings is zero. Discuss whether this is an outlier in context and how you would decide to keep or remove it.

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Question 10

Person A collected data on weekly junk food servings and average idle heart rate for $20$ weeks, but weeks $5$ , $12$ , and $17$ have missing heart rate readings. Calculate the number of complete weeks that remain for analysis.

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Question 11

The relationship between the number of servings of junk food consumed per week ( $x$ ) and the average idle heart rate ( $y$ ) in beats per minute (bpm) is modeled by a linear regression equation.

Using the regression line $y = 0.83x + 66.54$ , predict the average idle heart rate for a subject consuming $10$ servings of junk food in a week.

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Question 12

To smooth weekly junk food servings, Person A applies a 3-week moving average on the series $[3, 5, 4, 6, 7, 5, 4, 8, 2, 6]$ . Calculate the smoothed values for weeks 2–9.

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Question Type 4: Choosing the appropriate data given contextual information Exercises