CONCEPTO PESOS

Return to the hypothesis about men’s and women’s smoking. Smoking is the single most preventa- ble health risk factor in all societies. People have been smoking for centuries. By the mid-1960s it was clear that smoking had numerous adverse public health impacts. Public health initiatives have sought to eliminate, or at least reduce, people’s smoking.

Smoking depends on many factors. Younger people often start in order to look more mature. People living in certain cultures (often where tobacco is raised) smoke more, and men have

H1:Er 7 Ep

H0:Er = Ep

H1:cm Z cw

traditionally smoked more than women (although women appear to be catching up). People with more education may better recognize the health impacts of smoking, and hence smoke less. Economists hypothesize that, like most goods, smoking is negatively related to cigarette price (as price increases, smoking decreases). The impact of income is unclear.

To compare men’s and women’s smoking rates, we can sample the population, yet we know that there are lots of different types of people, and we would like to avoid the confounding influ- ences of age, education, or location. We could attempt to avoid this sort of distortion by drawing samples randomly from among all possible 20-year-olds, called the universe of data. Alternatively, we may try to choose samples of 20-year-old men and women from the same general income group. A sample of college sophomores, for example, from the same location and with similar socioeco- nomic status, may be a good group for holding many factors constant. Even this example shows how difficult things may be to control. People of the same age (at the same college) may come from different locations and different types of families. Some may have parents or other family members who smoke.

We need a test to determine the differences between two distributions of continuous data. Continuous data are natural measures that in principle could take on different values for each observation. Examples include height, weight, income, or price. Categorical data refer to arbitrary categories such as gender (male or female), race (black, white, or other), or location (urban or rural). In this chapter, unless we specify categorical data, our methods will refer exclusively to the analysis of continuous data.

Lots of people do not smoke, so let us concentrate only on smokers (looking at the decision whether or not to smoke is an important policy question, but is far beyond the scope of this example). The econometrician asks one woman and finds that she smokes 10 cigarettes per day (cw=10). The

first man asked smokes 15 cigarettes per day (cm=15). This provides evidence that men smoke more

than women, because cm7 cw, or 15 7 10. It is not very convincing evidence, however. The man, or

woman, or either, may not be typical of the entire group. What if a different man and/or woman had been selected? Would the answer have been different?

It seems logical to test several men and to compute the mean or average level by summing the levels and dividing them by the total number of men tested. The National Institutes of Health (in 2001 and 2002) collected a database of over 43,000 individuals called the National Epidemiologic Survey on Alcohol and Related Conditions, or NESARC. They focused on potentially substance abusive activities including smoking, drinking of alcoholic beverages, and the taking of recreational (and harder) drugs. They asked a number of questions about smoking, and from the analysis of smok- ers the textbook authors found that

For 4,714 men, the mean, or average level, was 15.60 cigarettes per day. For 4,841 women, the average level, was 13.47 cigarettes per day. The difference, d= then, is 2.13 cigarettes per day.

The Variance of a Distribution

Although a difference of the two means is improved evidence, the econometrician desires a more rigorous criterion. It could be that the true level for both men and women is 14 cigarettes per day, but our sample randomly drew a higher average level for men (15.60) than for women (13.47). Figure 3-1 plots the distributions in percentage terms. Almost 25 percent of the women smoke between 1 and 5 cigarettes per day, compared to a slightly smaller percentage of men; in contrast, while about 28 percent of the women smoke between 16 and 20 cigarettes per day, about 33 percent of the men smoke at this level. Although the mean levels differ (men are higher than women), some groups of women (those who smoke less) have higher percentages than some groups of men. Statisticians have found the variance of a distribution to be a useful way to summarize its disper- sion. To calculate the variance of women’s levels, we subtract each observation from the mean

cm - cw,

cw, cm,

0 5 10 15 20 25 30 35 1–5 6–10 11–15 16–20 21–30 31–40 +40 Cigarettes per Day

P

ercentage

Women Men

FIGURE 3-1 Cigarette Consumption by Gender, 2001–2002

(13.47), square that term, sum the total, and divide that total by the number of observations, N. Hence, variance, Vw, equals:

(3.5)

Here N1is the number of women who smoke 1 cigarette per day, N2is the number who smoke 2

cigarettes per day, and so on (and yes, there are some women who smoke 80 cigarettes per day!).

Vwreflects the variance of any individual term in the distribution. If V is large, then the

dispersion around the mean is wide and another woman tested might be far from our mean. If V is small, then the dispersion around the mean is narrow and another observation might be close to the mean.

Standard Error of the Mean

The variance is often deflated by taking the square root to get the standard deviation, s, yielding:

(3.6)

As with V, a large (small) value of s indicates a large (small) dispersion around the mean. Statisticians have shown that we can calculate the standard error of the mean itself by dividing s by the square root of the number of observations. In this sample, the standard deviation of the distribution for women equals 9.71. The standard error of the mean of the women’s distribution would then equal swdivided

by the square root of 4,841, or (9.71 , 69.6), which equals 0.14.1

A powerful theorem in statistics, the Central Limit Theorem, states that no matter what the underlying distribution, the means of that distribution are distributed like a normal, or bell-shaped, curve. Hence, we can plot the normal distribution of means of women’s levels with a mean of 13.47 and a standard error of 0.14.

sw = _EN1x11 - 13.472 2 _{+ N} 2x12 - 13.4722 + Á + N80x180 - 13.4722 4,841 Vw = N1x11 - 13.472 2 _{+ N} 2x12 - 13.4722 + Á + N80x180 - 13.4722 4,841

1_{Formally, in a sample (as opposed to the entire population), we calculate the standard error by dividing by n - 1. All}

Statisticians have also shown that a little more than 68 percent of the area under the curve would be within one standard error, or between levels of 13.33 (that is, 13.47 - 0.14) and 13.61 (i.e., 13.47 + 0.14), and that 95.4 percent would be within two standard errors. This means that we could be about 95 percent sure that the true mean quantity of cigarettes smoked for women was between 13.19, [that is, 13.47- (2 × 0.14)], and 13.75, [i.e., 13.47 + (2 × 0.14)]. A similar calculation can be done for men, yielding a similar measurement. Intuitively, the further apart the means and the smaller the dispersions (standard errors), the more likely we are to determine that the average level for men is smaller than that for women. To test the hypothesis formally, we then construct a “difference of means” test. We wish to compare the measurement d= to zero, which was the original hypothesis.

Here d 2.13. The variance of the difference is defined as the sum of the variances of the stan- dard errors. If the standard error for women was 0.14 as we calculated it and the standard error for men given the sample in Figure 3-2A was 0.17, then the standard error of the difference would be:

(3.7)

sd = 20.142 + 0.172 = 0.216

= cm - cw,

Distribution for Females and Males

Females 0.000 0.500 1.000 1.500 2.000 2.500 3.000 13.0 13.5 14.0 14.5 15.0 15.5 16.0 16.5 Cigarettes per Day

Probability Density Males 0.000 0.200 0.400 0.600 0.800 1.000 1.200 1.400 1.600 1.800 2.000 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 Difference in Level (Male—Female)

Probability Density

Difference of Means

The difference and its distribution also can be plotted.

The most probable value of the difference, as noted in Figure 3-2B, is 2.13. About 68 per- cent of the distribution lies between 1.91 (i.e., 2.13 - 1 × 0.216) and 2.35 (i.e., 2.13 + 1 × 0.216). About 95.4 percent of the distribution lies between 1.69 (i.e., 2.13 - 2 × 0.216) and 2.56 (i.e., 2.13+ 2 × 0.216).

This experiment would find very good evidence that among smokers, women smoke fewer cigarettes than men. The males have higher levels than the females, and the probability is well over 95 percent that this difference is statistically significant.

Alternatively, the t-statistic, comparing the numbers 2.13 and 0.0, equals 2.13 , 0.216, or ap- proximately 9.86. Statisticians calculate tables of t-statistics, whose critical values are related to the size of the sample. With a sample of over 8,000, a t-statistic of nearly 10 is statistically signif- icant at well over the 99 percent level. In other words, we can be 99+ percent certain that men smoke more cigarettes than women.

Hypotheses and Inferences

This process illustrates the steps that are necessary to test hypotheses appropriately. The econome- trician must:

1. State the hypotheses clearly

2. Choose a sample that is suitable to the task of testing.

3. Calculate the appropriate measures of central tendency and dispersion: the mean and the stan-

dard error of the mean for both men and women, leading to the difference of the two means.

4. Draw the appropriate inferences: men smoke more than women.

No matter how sophisticated the method used, good statistical analysis depends on the ability to address these four criteria and stands (or falls) on the success in fulfilling them. Box 3-1 provides a particularly good example of how analysts have examined the impacts of electromagnetic fields (EMFs) on children living near power lines.

There are, of course, measures of central tendency other than the mean (or average). Someone who smokes 4 packs per day (80 cigarettes) may unduly influence the mean. A different measure, the median, calculates a statistic such that half of the observations are greater than the median and half are less. Thus, a median smoking level of 15 cigarettes would imply that half of the people smoked more than 15, and half smoked fewer. The median is less sensitive to extreme values in the data (e.g., someone who smokes 60 cigarettes per day would have no more effect on a median of 15 than does someone who smokes 20). However, the median can present mathemati- cal problems in hypothesis testing. Simple formulas for standard errors of medians have not been available, although popular numerical “bootstrapping” methods now provide intuitive and accurate standard errors. For a good discussion of bootstrapping, see Efron and Tibshirani (1993).

H1:cm Z cw.

H0:cm = cw, against

BOX 3-1

No Link Between Childhood Cancer and Electromagnetic Fields

In 1979, Dr. Nancy Wertheimer and her assistant, Ed Leeper, reported that children who lived near power lines had twice the normal incidence of leukemia. The study was criticized because it was small and relied on indirect evidence rather than direct measurements of exposure to electromagnetic fields (EMFs). However, the report and its findings had major impacts. Parents of children with cancer sued power compa- nies. Owners near power lines saw reductions in the values of their homes.

2_{Although difference of means considers only two categories, analysis of variance methods allow the consideration of three}

or more categories. Newbold, Carlson, and Thorne (2007) present good discussions on this and other statistical topics.