1. Let the observed value of the mean X of a random sample of size 20 from a distribution that is N(.80) be 81.2. Find a 95% confidence interval for u. 2. Let X, X2,..., X,, be a random sample from a gamma distribution with a = 3 and unknown > 0. Derive a 100(1-)% confidence interval for . 3. The brightness of a television picture tube can be evaluated by measuring the amount of current required to achieve a particular brightness level. A sample of 10 tubes results in x = 317.2 and s=15.7. Find (in microamps) a 99% confidence interval on mean current required. State any necessary assumptions about the underlying distribution of the data. 4. A pharmaceutical company makes tranquilizers. It is assumed that the distribution for the length of time they last is approximately normal. Researchers in a hospital used the drug on a random sample of nine patients. The effective period of the tranquilizer for each patient (in hours) was as follows: 2.7; 2.8; 3.0; 2.3; 2.3; 2.2; 2.8; 2.1; and 2.4. Find a 95% confidence interval for the population mean length of time.