35.13 Exercises
Selected answers are available in Sect. D.32.
Exercise 35.1 In wastewater treatment facilities, air from biofiltration is passed through a membrane and dissolved in water, and is transformed into harmless byproducts (Chitwood and Devinny 2001; Devore and Berk 2007). The removal efficiency y (in %) may depend on the inlet temperature (in ∘C; x). The RQ is
In treating biofiltation wastewater, how does the removal efficiency depend on the inlet temperature?
Using the scatterplot (Fig. 35.20) and software output (Fig. 35.21 (jamovi); Fig. 35.22 (SPSS)), answer these questions.
- Write down the value of the slope (b1) and y-intercept (b0).
- Write down the regression equation.
- Interpret the slope (b1).
- Test for a relationship in the population, by first writing the hypotheses.
- Write down the t-score and P-value from the software output.
- Determine an approximate 95% CI for the population slope β1.
- Write a conclusion.

FIGURE 35.20: The relationship between removal efficiency and inlet temperature

FIGURE 35.21: jamovi regression output for the removal-efficiency data
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FIGURE 35.22: SPSS regression output for the removal-efficiency data
Exercise 35.2 A study (Myers (1990), p. 75) of American footballers measured the right-leg strengths x of 13 players (using a weight lifting test), and the distance y they punt a football (with their right leg).
- Use the plot (Fig. 35.23) to guess of the values of the intercept and slope.
- Using the jamovi output (Fig. 35.24), write down the value of the slope (b1) and y-intercept (b0).
- Hence write down the regression equation.
- Interpret the slope (b1).
- Write the hypotheses for testing for a relationship in the population
- Write down the t-score and P-value from the output.
- Determine an approximate 95% CI for the population slope β1.
- Write a conclusion.

FIGURE 35.23: Punting distance and right leg strength

FIGURE 35.24: jamovi regression output for the punting data
Exercise 35.3 A study (Amin and Mahmood-ul-Hasan 2019) of gas engines measured the throttle angle (x) and the manifold air pressure (y) as a fraction of the maximum value.
- The value of r is given in the paper as 0.972986604. Comment on this value, and what it means.
- Comment on the use of a regression model, based on the scatterplot (Fig. 35.25).
- The authors fitted the following regression model: y=0.009+0.458x. Identify errors that the researchers have made when giving this regression equation (there are more than one).
- Critique the researchers’ approach.

FIGURE 35.25: Manifold air pressure plotted against throttle angle for an internal-combustion gas engine
Exercise 35.4 A study of hot mix asphalt (Panda et al. 2018) created n=42 samples of asphalt and measured the volume of air voids and the bitumen content by weight (Fig. 35.26). Use the software output (Fig. 35.27) to answer these questions.
- Write down the regression equation.
- Interpret what the regression equation means.
- Perform a test to determine is there is a relationship between the variables.
- Predict the mean percentage of air voids by volume when the percentage bitumen is 5.0%. Do you expect this to be a good prediction? Why or why not?
- Predict the mean percentage of air voids by volume when the percentage bitumen is 6.0%. Do you expect this to be a good prediction? Why or why not?

FIGURE 35.26: Air voids in bitumen samples

FIGURE 35.27: jamovi regression output for the bitumen data
Exercise 35.5 A study of n=31 people on the use of sunscreen (Heerfordt et al. 2018) explored the relationship between the time (in minutes) spent on sunscreen application x and the amount (in grams) of sunscreen applied (y). The fitted regression equation was ˆy=0.27+2.21x.
- Interpret the meaning of b0 and b1. Do they seems sensible?
- According to the article, a hypothesis for testing β0 produced a P-value much larger than 0.05. What does this mean?
- If someone spent 8 minutes applying sunscreen, how much sunscreen would you predict that they used?
- The article reports that R2=0.64. Interpret this value.
- What is the value of the correlation coefficient?
Exercise 35.6 One study (Bhargava et al. 1985) stated:
In developing countries […] logistic problems prevent the weighing of every newborn child. A study was performed to see whether other simpler measurements could be substituted for weight to identify neonates of low birth weight and those at risk.
— Bhargava et al. (1985), p. 1617
The relationship between infant chest circumference (in cm) x and birth weight (in grams) y was given as:
ˆy=−3440.2403+199.2987x. The correlation coefficient was r=0.8696 with P<0.001.
- Based on the regression equation only, could chest circumference be used as a useful predictor of birth weight? Explain.
- Based on the correlation information only, could chest circumference be used as a useful predictor of birth weight? Explain.
- Interpret the intercept and the slope of the regression equation.
- What units of measurement are the intercept and slope measured in?
- Predict the birth weight of an infant with a chest circumference of 30cm.
- Critique the way in which the regression equation and correlation coefficient are reported.