Suppose we have the following two sample data where the two samples have the same size(Table 2). index | 1 2 12 sample 1 3.00 1.92 1.46 sample 2 1.60 -2.36 2.32 Table 2: Two sample data Let's write sample 1 as x = ($1,222, - -- ,2312) and sample 2 as y = (91,99, - -- ,y12) (e.g. $1 = 3.00 and y1 = 1.60). x and y satisfy a = 1.0,g = 0.43,s=1.30,s= 2.49 Now we want to test if the two population means are equal, i.e. we test Hazp1=p2vsH1:p17p2. (a) Suppose the two samples are from independent normal distributions with equal variances (unknown). In other words, our model is (i) (3pt) Write the test statistic and its distribution under H0. (ii) (4pt) Compute the value of the realized test statistic. (iii) (4pt) Compute the p value. What do you conclude at level or = 0.05? (b) Now assume that this is a paired data. Let d1: = mg y; be the difference of the ith observations and assume that the differences follow a normal distribution, i.e., D1: D2! ' ' ' 54012 1:81 All(#4030123) where p9 = #1 p2 and 0% is an unknown positive constant. The sample variance of dg's is given by 8% = 0.20. (i) (3pt) Write the test statistic and its distribution under H0. (ii) (4pt) Compute the value of the realized test statistic. (iii) (4pt) Compute the p value. What do you conclude at level or = 0.05