LESSON 2 • Least Squares Regression and Correlation 299

IInvest invest iggationation 44 Association and Causation Association and CausationReports in the media often suggest that research has found a cause-and-effect relationship between two variables. For example, a newspaper article listed several “weird” things that are associated with whether or not a student graduates from college. The following excerpt is about one of them.

Five Weird Ways to College Success

Don’t smoke.[Alexander] Astin and [Leticia] Oseguera [of UCLA] examined the graduation rates of 56,818 students at 262 colleges, a huge sample, and reported that smoking had one of the largest negative associations with degree completion.

Source: Jay Mathews, The Washington Post, June 13, 2006, www.washingtonpost.com/wp-dyn/content/article/2006/06/13/AR2006061300628.html

As you work on the problems in this investigation, look for answers to the following question:

When you have an association between two variables, how can you determine if the association is a result

of a cause-and-effect relationship?

1 The 12 countries listed below have the highest per person ice cream consumption of any countries in the world. As shown in the following table and scatterplot, there is an association between the number of recorded crimes and ice cream consumption.

Country Ice Cream Consumption per Person (in liters) per Year

Recorded Crimes per 100,000 Inhabitants per Year

New Zealand 26.3 12,591

United States 22.5 9,622

Canada 17.8 8,705

Australia 17.8 6,161

Switzerland 14.4 4,769

Sweden 14.2 13,516

Finland 13.9 7,273

Denmark 9.2 1,051

Italy 8.2 4,243

France 5.4 6,765

Germany 3.8 8,025

China 1.8 131

Name:_____________________________________

Unit 3Lesson 2

Investigation 4

300 UNIT 4 • Regression and Correlation

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Source: United Nations Office on Drugs and Crime www.unodc.org/unodc/crime_cicp_surveys.html and www.foodsci.uoguelph.ca/dairyedu/icdata.html. Their source: The Latest Scoop, 2000 Edition, Int. Dairy Foods Assn.

a. For the data above, the regression line is y = 343x + 2,500, and the correlation is 0.637. Interpret the slope of the regression line in the context of this situation.

b. Do these data imply that if a country wants to decrease the crime rate, it should ask people to eat less ice cream? Explain your reasoning.

c. The following scatterplot shows the variables reversed on the axes. The regression equation is now y = 0.00118x + 4.77. Interpret the slope of this regression line. Can you now say that if the crime rate increases, then people will eat more ice cream?

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Recorded Crimes per 100,000 Inhabitants0 4,000 8,000 12,000

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d. A lurking variable is a variable that lurks in the background and affects both of the original variables. What are some possible lurking variables that might explain this association between crime rate and consumption of ice cream?

There are several reasons why two variables may be correlated:

• The two variables, A and B, have a cause-and-effect relationship. That is, an increase in the value of A, called the explanatory (or independent) variable, tends to cause an increase (or decrease) in the value of B called the response (or dependent) variable.

A B

LESSON 2 • Least Squares Regression and Correlation 301

• The two variables, A and B, have nothing directly to do with each other. However, an increase in the value of a lurking variable C tends to cause the values of each of the two variables to increase together, to decrease together, or one to increase and the other to decrease.

C

A B

• Even though the correlation between the two variables is actually zero or close to zero in the entire population, you get a nonzero correlation just by chance when you have a small number of observations.

A B

2 The association for each pair of variables below is strong. Decide which of the three reasons above best explains the association. Then, where appropriate, draw a directed graph to indicate the relationship between the two variables.

a. Number of hours of studying each week and GPA

b. Reading ability of a child and his or her shoe size

c. Value of a car and its age

d. Degree of baldness of a man and probability of a heart attack in the next year

e. The median household income in the U.S. and skin cancer rate over the years

f. Number of people attending a movie and income from ticket sales

g. Number of letters in first name and age for a group of three adult women

3 Suppose you ask everyone in your community who has a phone for the number of letters in his or her last name and for the sum of the last four digits of his or her phone number.

a. Should there be positive association, negative association, or no association? Why?

b. Collect the information above from five members of your class or from a local telephone book, and compute the correlation r between number of letters in last name and sum of last four digits of phone number.

c. Did you get the correlation that you predicted in Part a? Explain why this should or should not be the case.

4 Look back at the article excerpt from the Washington Post on page 299.

a. What variables are said to be associated in the article? Which is considered the explanatory variable and which is the response variable?

b. What are some possible reasons for the association that are not mentioned in the article?

302 UNIT 4 • Regression and Correlation

c. Answer the questions in Parts a and b for the following Los Angeles Times article.

Tall Men Display Greater Risk of Skin Disease

A poll conducted by University of Washington researchers in Seattle found

that men taller than 6 feet, 1 inch had almost 2 1 _ 2 times the risk of developing

melanoma, an often fatal form of skin cancer, as those who were shorter than 5-foot-8.

Source: Los Angeles Times, January 14, 2002, page S2.

5 In the following study, the researchers tried to “control” for lurking variables by taking them into account.

Mind Games May Keep the Brain Sharp

An absorbing book or a challenging crossword puzzle may keep your mind more than busy. It may keep it healthy, too, according to a 21-year study of mental breakdown in old age. …

In the Einstein College study of 469 elderly people, those in the top third in mental activity had a 63 percent lower risk of dementia than the bottom third. Taking part in a single activity one day a week reduced the risk by 7 percent.

The use-it-or-lose-it notion is not a new idea. Other researchers have discovered evidence that mental activity may guard against dementia. But it is hard to prove since early dementia without obvious symptoms may cause people to slack off their hobbies. If this is so, dementia affects hobbies—and maybe not the reverse.

The researchers tried to minimize that possibility by considering only those who were dementia-free for seven years after joining the study. They also tried to eliminate the potential role of education and intelligence in guarding against dementia.

The study also took physical exercise into account. Nearly all physical activities, including stair climbing and group exercises, appeared to offer no protection against dementia. The only exception was frequent dancing, perhaps because dance music engages the dancer’s mind, suggested lead researcher Joe Verghese, a neurologist at Einstein College.

Source: www.cnn.com/2003/HEALTH/conditions/06/19/avoiding.dementia.ap/index.html

a. Identify the explanatory and response variables.

b. Name the lurking variables that the researchers considered.

c. Describe how the lurking variables might be taken into consideration.

d. Give at least one other possible lurking variable that is not mentioned.

LESSON 2 • Least Squares Regression and Correlation 303

Summarize the Mathematics

6 By now, you may be wondering how anyone could ever know whether an association means that one variable causes the other or whether there is a lurking variable that causes both. The only way to find out for sure is to conduct an experiment. In an experiment, volunteer subjects are randomly assigned to two or more different treatments. For example, suppose you want to decide if a cup of tea causes reduction in pain from a tension headache. You cannot just give a cup of tea to people with a tension headache and see if it goes away because some headaches go away over time without any treatment at all. So, you randomly divide your group of volunteers into those who get a cup of tea and those who get a cup of hot water. By randomizing, you hope to balance the people whose headaches go away quickly without any treatment at all between the two treatment groups. So if the headaches of those with tea tend to go away more quickly, you will know that it is the tea that caused it, not just sitting awhile to have a drink or not just the extra hot liquid.

a. For the study “Mind Games May Keep the Brain Sharp” from Problem 5, describe how you could conduct an experiment to decide whether one variable actually causes the other.

b. For each of these studies, explain why it is impossible to do an experiment to determine cause-and-effect.

i. “Five Weird Ways to College Success” from page 299

ii. “Tall Men Display Greater Risk of Skin Disease” from Problem 4 Part c

In this investigation, you learned how to distinguish between correlation and cause-and-effect in situations that involved an association between two variables.a Describe a situation involving two variables for which the correlation is strong, but there

is no cause-and-effect relationship.

b Describe a situation involving two variables for which the correlation is strong and where a change in one variable causes a change in the other variable.

c Explain what is meant by the often-repeated statement, “Correlation does not imply causation.”

d When you make a scatterplot, on which axis should you put the explanatory variable?

e How can you be certain whether an association means that there is a cause-and-effect relationship between two variables?

Be prepared to share your ideas and examples with the class.