Homework

1. Read chapter 7, through page 453.  (Two hours.)
2. (Example 5.16 of the text.)  Compute a 95% confidence interval for the mean of the reference area TcCB data.  To document the computation, create a small table showing the estimated mean, standard deviation, standard error, t-statistic, and confidence limits.  Use the same approach with the following problems.
3. (Example 5.17).  Compute a 95% UCL for the variance of the log-transformed Reference Area TcCB data.  Compute a 95% UCL for the geometric mean of the original TcCB data.
4. (Examples 5.19 and 5.21).  Compute upper and lower 95% confidence limits for the mean of the chromium data using the Land and Normal theory methods.  Which method is more appropriate to use?  Why?
5. (Examples 5.24 and 6.6).  Verify the computations in Table 5.15 of the text.  It displays the mean, SD, 95th percentile, and 99% LCL on the 95th percentile by well for the aldicarb concentrations.  Add a row for the 95% LCL on the 99th percentile.  Do these equal the 99% LCLs on the 95th percentile?
6. (Example 5.24, continued).  Reproduce the results concerning the two-sided 99% confidence intervals for the 95th percentiles at each well.
7. (Example 5.25).  Reproduce Table 5.17 of the text.  It displays the mean, CV, 95th percentile, and LCL on the 95th percentile by well for the chrysene concentrations using a lognormal distribution assumption.  Create a comparable table using a normal assumption.  How do these tables compare?
8. (Example 5.25, continued).  Reproduce the two-sided 99% confidence intervals for the 95th percentiles at each well, using both the lognormal and normal distributional assumptions.  How do the results compare?
9. (Example 6.1).  Using the Bonferroni approximation, compute the one-sided upper 95% prediction limit for the next four observations of arsenic.  (*) Explore how the prediction limit changes as you increase k=4 to k=5, 6, ... etc.  Does it approach some maximum value as k grows without bound?  (You will probably overflow the computer or calculator in this exercise, so you need to think about what a prediction limit means in order to arrive at a defensible answer.)
10. (Example 6.7).  Compute a one-sided upper 95%-coverage, 95%-confidence tolerance limit for the Reference Area TcCB concentrations using both normal and lognormal assumptions.  How do the limits compare?  Which assumption is more appropriate to use? 
11. (Example 6.7, continued).  Compare the Cleanup Area observations to these two limits.  (*) Use the binomial distribution to evaluate whether too much (i.e., more than five percent) of the Cleanup Area concentrations exceed the tolerance limits.
12. (Example 6.7, continued).  Compute 95%-confidence upper prediction limits to contain the next 77 observations using both normal and lognormal assumptions.  Compare the 77 Cleanup Area observations to these two limits (lognormal and normal).

Each of problems 2-12 should take only a few minutes each, once you have acquired appropriate software, have learned to use it, and have the data in computerized form.  The problems involving lognormal assumptions will take more time because you will have to compute logarithms of the data, determine the appropriate procedures, and possibly back-transform the results.

Although these problems look like "busy work," that is only because the answers already appear in the text.  It is important to be able to produce correct answers when you are analyzing your own data.  One of the few ways to ensure your calculations are correct is to establish that you can reproduce published results.  Therefore these problems consist of work that you would have to do anyway if you ever want to compute confidence, prediction, or tolerance limits using the Normal or Lognormal assumptions.