Analyzing Data, the Interpreting the Analyses, and Communicating the Results (4). A paired $t$ test for the difference data in Exercise 6.67 is shown here.
The $p$-value in the output reads .000 , which means that it is smaller than .0005 ( 1 chance in 2,000 ). Thus, it is extremely unlikely that we would see data as extreme as those actually collected unless workers at the battery factory were contaminating their children. We reject the null hypothesis and conclude that the difference between the lead levels of children in the exposed and control groups is large enough to be statistically significant.
The next question is whether the difference between the two groups is large enough to be of practical importance. This is a judgment for people who know about lead poisoning to make, not for statisticians. The best estimate of the true (population) mean difference is 15.97 , or about 16. On average, children of workers in the battery plant have about $16 \mu \mathrm{g} / \mathrm{dl}$ more lead than their peers whose parents do not work in a lead-related industry. Almost any toxicologist would deem this increase to be dangerous and unacceptable. (The mean of the control group is also about 16. On average, the effect of having a parent who works in the battery factory is to double the lead level. Doubling the lead level brings the average value for exposed children to about 32 , which is getting close to the level where medical treatment is required. Also remember that some toxicologists believe that any amount of lead is harmful to the neurological development of children.)
a. Should the $t$ test we did have been one-sided? In practice, we must make the decision to do a one-sided test before the data are collected. We might argue that having a parent working at the battery factory could not decrease a child's exposure to lead.
1) Write the null hypothesis and its one-sided alternative in both words and symbols. Perform the test. How is its p-value related to the p-value for the two-sided test?
2) It might be tempting to argue that children of workers at a lead-using factory could not have generally lower levels of lead than children in the rest of the population. But can you imagine a scenario in which the mean levels would really be lower for exposed children?
b. We used a $t$ test to confirm our impression that exposed children have more lead in their blood than their control counterparts. Although there is no clear reason to prefer nonparametric tests for these data, verify that they yield the same conclusion as the $t$ test does.