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Describe an association between two variables. Give a confounding variable that may help to account for this association. People with shorter hair tend to be taller.

To examine whether eating brown rice affects metabolism, we ask a random sample of people whether they eat brown rice and we also measure their metabolism rate.

To examine whether planting trees reduces air pollution, we find a sample of city blocks with similar levels of air pollution and we then plant trees in half of the blocks in the sample. After waiting an appropriate amount of time, we measure air pollution levels.

Three situations are described at the start of this section, on page \(29 .\) In the third bullet, we describe an association between the amount of salt spread on the roads and the number of accidents. Describe a possible confounding variable and explain how it fits the definition of a confounding variable.

In elementary school (grades 1 to 6 ), there is a strong association between a child's height and the child's reading ability. Taller children tend to be able to read at a higher level. However, there is a very significant confounding variable that is influencing both height and reading ability. What is it?

In 2008 , a study \(^{46}\) was conducted measuring the impact that music volume has on beer consumption. The researchers went into bars, controlled the music volume, and measured how much beer was consumed. The article states that "the sound level of the environmental music was manipulated according to a randomization scheme." It was found that louder music corresponds to more beer consumption. Does this provide evidence that louder music causes people to drink more beer? Why or why not?

"Antibiotics in infancy may cause obesity in adults," claims a recent headline. \(^{49}\) A study in mice randomly assigned infant mice to either be given antibiotics or not, and the mice given antibiotics were more likely to be obese as adults. A separate study in humans found that children who had been given antibiotics before they were a year old (for example, for an ear infection) were more likely to be obese as adults. (Researchers believe the effect may be due to changes in the gut microbiome.) Based on these studies, is the headline an appropriate conclusion to make: (a) For mice? (b) For humans?

Infections Can Lower IQ A headline in June 2015 proclaims "Infections can lower IQ." 1 The headline is based on a study in which scientists gave an IQ test to Danish men at age \(19 .\) They also analyzed the hospital records of the men and found that \(35 \%\) of them had been in a hospital with an infection such as an STI or a urinary tract infection. The average IQ score was lower for the men who had an infection than for the men who hadn't. (a) What are the cases in this study? (b) What is the explanatory variable? Is it categorical or quantitative? (c) What is the response variable? Is it categorical or quantitative? (d) Does the headline imply causation? (e) Is the study an experiment or an observational study? (f) Is it appropriate to conclude causation in this case?

A 2014 headline reads "Sitting Is the New Smoking: Ways a Sedentary Lifestyle is Killing You," 12 and explains the mounting evidence for ways in which sitting is bad for you. A more recent large 2015 study \(^{53}\) contributed to this evidence by following 69,260 men and 77,462 women and finding that for women, those who spent more leisure time sitting were significantly more likely to get cancer. (a) What are the explanatory and response variables for the 2015 study? (b) Is the 2015 study an observational study or a randomized experiment? (c) Can we conclude from the 2015 study that spending more leisure time sitting causes cancer in women? Why or why not? (d) Can we conclude from the 2015 study that spending more leisure time sitting does not cause cancer in women?

It is well-known that lack of sleep impairs concentration and alertness, and this might be due partly to late night food consumption. A 2015 study \(^{54}\) took 44 people aged 21 to 50 and gave them unlimited access to food and drink during the day, but allowed them only 4 hours of sleep per night for three consecutive nights. On the fourth night, all participants again had to stay up until 4 am, but this time participants were randomized into two groups; one group was only given access to water from \(10 \mathrm{pm}\) until their bedtime at \(4 \mathrm{am}\) while the other group still had unlimited access to food and drink for all hours. The group forced to fast from \(10 \mathrm{pm}\) on performed significantly better on tests of reaction time and had fewer attention lapses than the group with access to late night food. (a) What are the explanatory and response variables? (b) Is this an observational study or a randomized experiment? (c) Can we conclude that eating late at night worsens some of the typical effects of sleep deprivation (reaction time and attention lapses)?