1. we would like to compare the mea

1. we would like to compare the means of log-rainfull from the seeded and unseeded clouds that were discussed originally in Example 7.3.1. Let.........be the log-rainfulls of the 26 seeded clouds, and let..... be the log-rainfulls of 26 unseeded clouds. The observed values of sample statistics are.....and....The null hypothesis is that seeded clouds produce no more rainfull than unseeded clouds, while the alternative hypothesis is that seeded clouds produce more rainfull. Test these hypotheses at the level of significance ........., assuming that the variances of all log-rainfulls are the same

2. Suppose that a certain drug A was administered to eight patients selected at random, and after a fixed time period, the concentration of the drug in certain body cells of each patient was measu red in appropriate units. Suppose that these concentrations for the eight patients were found to be as follows:
.......................
Suppose also that a second drug B was administered to six different patients selected at random,and when the concentration of drug B was measured in a similar way for these six patients , the results were as follows:
...........................
Assuming that all the observations have a normal distribution with a common unknown variance, test the following hypotheses at the level of significance 0.10: The null hypothesis is that the mean concentration of drug A among all patients is at least as large as the mean concentration of drug B. The alternative hypothesis is that the mean concentration of drug B is larger then that of drug A.
3. Consider again the conditions of Exercise 2, but suppose now that it is desired to test the following hypotheses: The null hypothesis is that the mean concentration of drug A among all patients is the same as the mean concentration of drug B. The alternative hypothesis, wnich is two-sided, is that the mean concentrations of the two drugs are not the same. Find the number c so that the level 0.05 two-sided t test will reject.........,where U defined by Eq. (8.6.3). Also, perform the test.
4. Suppose that....... form a random sample from a normal distribution with mean....and variance .....and...... form an independent random sample from a normal distribution with mean....and variance...show that if.....and..... then the random variable U defined by Eq.(8.6.9) has a t distribution with...... degrees of freedom.