Test Your Understanding 5

It has been reported (The Boston Globe, July 31, 1997) that as much as
25% of all worker's compensation claims and 40% of all
payments are attributable to lower back injuries. In an effort
to reduce such injuries, employers have sent workers ``back to
school'' to learn to lift heavy objects safely. Suppose that,
in order to study the effectiveness of such a program, for one
year after the program's completion, you monitor the incidence
of back injury for a random sample of 200 workers who have
taken the program, and a random sample of 200 workers who have
not. Of the 200 workers who completed the course, 18 report at
least one lower back injury during this period and of the
other group, 21 report at least one lower back injury. Is this
convincing evidence for the efficacy of the program? Formulate
the scientific and statistical hypotheses, the statistical
model, and the standardized test statistic. Obtain the
*p*-value and use it to state your conclusion.

SOLUTION:

*
*

- Scientific hypothesis: The program works.
- Statistical model: Two-sample binomial. Let
*Y*denote the number of injuries in the sample of course grads, and_{1}*Y*the number of injuries in the sample of non-grads, and assume that_{2}*Y*and_{1}*Y*are independent with and ._{2} - Statistical hypotheses:
*H*:_{0}*p*-_{1}*p*=0,_{2}*H*_{a}:*p*1-*p*2<0. -
*p*-value: First, compute the standardized test statistic, which under*H*has an approximate_{0}*N*(0,1) distribution provided the sample sizes are large enough. Here, is the pooled estimate of the common value,*p*, of*p*and_{1}*p*._{2}From the data, we have,

Therefore, the*y*=18,_{1}*n*-_{1}*y*=182,_{1}*y*=21, and_{2}*n*-_{2}*y*=179 are all larger than 10, so the normal approximation is ok. Also, the pooled estimate of_{2}*p*is . The observed value of the test statistic is*p*-value is (*Z*denotes a*N*(0,1) random variable) - Conclusion: Do not reject
*H*. The data supply insufficient evidence of the efficacy of the program._{0}