Air pollution is a serious problem in many places. One form of air pollution that is suspected to cause respiratory illness is particulate matter (PM), which consists of tiny particles in the air. Particulate matter can come from many sources, most commonly ash from burning, but also from other sources such as tiny particles of rubber that wear off of automobile and truck tires.
The town of Libby, Montana, was recently the focus of a study on the effect of PM on the respiratory health of children. Many houses in Libby are heated by wood stoves, which produce a lot of particulate pollution. The level of PM is greatest in the winter when more stoves are being used, and declines as the weather becomes warmer. The study attempted to determine whether higher levels of PM affect the respiratory health of children. In one part of the study, schoolchildren were given a questionnaire to bring home to their parents. Among other things, the questionnaire asked whether the child had experienced symptoms of wheezing during the past 60 days. Most parents returned the questionnaire within a couple of weeks. Parents who did not respond promptly were sent another copy of the questionnaire through the mail. Many of these parents responded to this mailed version.
Table 1.2 presents, for each day, the number of questionnaires that were returned by parents of children who wheezed, the number returned by those who did not wheeze, the average concentration of particulate matter in the atmosphere during the past 60 days (in units of micrograms per cubic meter), and whether the questionnaires were delivered in school or through the mail.
Table 1.2
Date PM Level Number of People Returning Questionnaires Number Who Wheezed School/Mail
March 5 19.815 3 0 School
March 6 19.885 72 9 School
March 7 20.006 69 5 School
March 8 19.758 30 1 School
March 9 19.827 44 7 School
March 10 19.686 31 1 School
March 11 19.823 38 3 School
March 12 19.697 66 5 School
March 13 19.505 42 4 School
March 14 19.359 31 1 School
March 15 19.348 19 4 School
March 16 19.318 3 1 School
March 17 19.124 2 0 School
April 12 14.422 10 1 Mail
April 13 14.418 9 1 Mail
April 14 14.405 8 0 Mail
April 15 14.141 3 0 Mail
April 16 13.910 4 0 Mail
April 17 13.951 2 0 Mail
April 18 13.545 2 0 Mail
April 20 13.326 3 0 Mail
April 22 13.154 2 0 Mail
We will consider a PM level of 17 or more to be high exposure, and a PM level of less than 17 to be low exposure.
- How many people had high exposure to PM?
- How many of the high-exposure people had wheeze symptoms?
- What percentage of the high-exposure people had wheeze symptoms?
- How many people had low exposure to PM?
- How many of the low-exposure people had wheeze symptoms?
- What percentage of the low-exposure people had wheeze symptoms?
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Is there a large difference between the percentage of high-exposure people with wheeze symptoms and the percentage of low-exposure people with wheeze symptoms? - Explain why the percentage of high-exposure people with wheeze symptoms is the same as the percentage of school-return people with wheeze symptoms.
- Explain why the percentage of low-exposure people with wheeze symptoms is the same as the percentage of mail-return people with wheeze symptoms.
- As the weather gets warmer, PM goes down because wood stoves are used less. Explain how this causes the mode of response (school or mail) to be related to PM.
- It is generally the case in epidemiologic studies that people who have symptoms are often eager to participate, while those who are unaffected are less interested. Explain how this may cause the mode of response (school or mail) to be related to the outcome.
- Rather than send out questionnaires, the investigators could have telephoned a random sample of people over a period of days. Explain how this might have reduced the confounding.
2 Case Study: Do Hybrid Cars Get Better Gas Mileage?
In the chapter introduction, we presented gas mileage data for 2016 model year hybrid and small non-hybrid cars. We will use histograms and back-to-back stem-and-leaf plots to compare the mileages between these two groups of cars. The following tables present the mileages, in miles per gallon.
Mileage Ratings for 2016 Hybrid Cars
42 31 50 25 31 32 20 30
26 20 31 28 42 21 50 34
29 30 41 29 40 20 42 26
46 29 28 42 40 33 42 28
28 30 37 26 37 44 40 42
50 26 45 41 52 40 38 37
30 21 40 56 34 47 40 26
Source: www.fueleconomy.gov
Mileage Ratings for 2016 Small Non-hybrid Cars
37 35 50 36 36 35 34 35
34 34 45 34 36 36 35 35
36 35 35 37 34 35 35 34
40 36 35 36 36 35 37 35
35 36 36 35 34 35 34 37
35 35 36 34 34 35 34 36
50 39 45 44 36 37 35 36
Source: www.fueleconomy.gov
- Construct a frequency distribution for the hybrid cars with a class width of 2.
- Explain why a class width of 2 is too narrow for these data.
- Construct a relative frequency distribution for the hybrid cars with a class width of 3, where the first class has a lower limit of 20.
- Construct a histogram based on this relative frequency distribution. Is the histogram unimodal or bimodal? Describe the skewness, if any, in these data.
- Construct a frequency distribution for the non-hybrid cars with an appropriate class width.
- Using this class width, construct a relative frequency distribution for the non-hybrid cars.
- Construct a histogram based on this relative frequency distribution. Is the histogram unimodal or bimodal? Describe the skewness, if any, in these data.
- Compare the histogram for the hybrid cars with the histogram for the non-hybrid cars. For which cars do the mileages vary more?
- Construct a back-to-back stem-and-leaf plot for these data, using two lines for each stem. Which do you think illustrates the comparison better, the histograms or the back-to-back stem-and-leaf plot? Why?
Sample Solution