1) The economics dept. at ISU conducts an annual survey of farmland value in the state of Iowa. In the Central region, which includes Ames and Story County, an acre of farmland this year has an average value of $4,529 ( 2007 Farmland Value Survey). This value is estimated from a survey, which we will assume is a simple random sample of N participants from a very large population, so the usual s.e. formula for the mean is appropriate.

The published material on the farmland survey includes very little about sample sizes and nothing about variability or precision of the results. Sample size calculations for surveys are usually based on the standard error of the mean or the width of a 95% confidence interval for the mean. For subsequent calculations, assume that the standard deviation (i.e. the variability between participants) is $750.

a) How many participants should you survey so that the s.e. of the mean equals $100?

b) How many participants should you survey so that the width of a 95% confidence interval for the mean equals $100.

2) Consider an experiment to evaluate whether an herbal extract affects the immune system. The response can be measured using a particular cell assay system (a very complicated procedure, but the details are not relevant here). There are two treatments in this experiment: extract and control. Each treatment will be randomly assigned to a tube of cells. You are contemplating an experiment with n=5 replicates of each treatment. The variability between replicates has been measured in other experiments: s.d. = 4.1. You don't know whether the herbal extract will increase or decrease the immune response, so you plan to use a 2-sided test.

a) What is the standard error of the difference between the two means? (Remember, there are n=5 replicates per treatment).
b) How many degrees of freedom are associated with the standard error (or equivalently, with the T statistic)? What is the 0.975 quantile of the appropriate T distribution?
c) If the true difference is 6.5, what is the power (at least approximately) of an alpha=5% two-sample t-test. Remember there are n=5 replicates per treatment, the population sd = 4.1, and you will use a 2-sided test.
d) Use the power approach to find the required number of replicates to provide 90% power for an alpha=5% test, when the true difference is 6.5. You will need to repeat the calculation to make sure you have reasonable quantiles. If you have reasonable starting values, you will only need to do the calculations twice. If your starting values are way off, you will need to do the calculations three times.