wuulru-ual 1- Let us take a sample from population X from a normal distribution with unknown mean #1- and unknown variance erg, and population Y from normal distribution with unknown mean ,ny and unknown variance o3. Suppose that we want to nd the interval estimation of dierences between two population means #1 uy. Let $1,332., - -- ,3\"! denote the random sample of size 'nI form the population X and let yl, yg, - - - , yr denote the random sample of size 11,, form the population Y. 2 2 Assume that the populations X and Y erg = cry = or . 2 We have used in our class the usual estimator of the common variance or is obtained by . . . . 2 _ pooling the sample data to obtain the pooled miance estimator sP. Eli-Elie - 573)2 + Slide;- a? 92: P nz+ny2 Let 83, and s: be the sample variances of sample X and sample Y respectively. a) Show that 32}: is an unbiased estimator for the common variance 0'2. 13) Show that if n3 = ny, s12: is simply the average of s: and 3:. c} Show that if T1,: a my and in.ac :5 way, 32;: is the weighted average of s: and s given to sample X. 2 y, with larger weight d} A study has been made to compare the nicotine contents of two brands of cigarettes. Ten cigarettes of Brand A has an average nicotine content of 3.1 milligrams with standard deviation of [1.5 milligrams, while eight cigarettes of Brand B had an average nicotine content fo 2.? milligrams with a standard deviation Uh? milligrams. Assuming that the two sets of data are independent random samples from normal populations with equal variances, construct a a 95% condence interval for p3,; p13, the true difference between the mean nicotine contents of the two brands of cigarettes. You may use ZTestU or t.test() to verify your answer, but replicate the values mathematically yourself and show your work as well. e) Repeat part d under the assumption that the variances of each brand are not equal. This time, you may use ZTe'stU or t.test{] on its own. Explain how this problem is dierent from part d, but no need to mathematically verify the values