3) 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 and wonder how many replicates are needed. 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) If you use n=5 replicates of each treatment, what is the standard error of the difference between the two means? 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) How different do the extract and control means need to be to have 80% power? Remember there are n=5 replicates per treatment, the population sd = 4.1, and you will use a 2-sided test with alpha = 0.05. d) Use the power approach to find the required number of replicates to provide 80% power for an alpha=5% test, when the true difference is 6.5? e) 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? Notes for d and e: 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.