Question

The difference of two independent normally distributed random variables is also normally distributed. We have used this fact in many of our derivations. Now, consider two independent and normally distributed populations with unknown variances σ 2 X and σ 2 Y . If we get a random sample X1, X2, . . . , Xn from the first population and a random sample Y1, Y2, . . . , Yn from the second population (note that both samples are of equal size n), then can you recommend a hypothesis test to accept or reject that the two populations have the same mean? Explain what the setup of the hypothesis test is, which statistic you would use, and when you would accept or reject the null hypothesis.

Question 4: Normally distributed populations The difference of two independent normally distributed random variables is also normally distributed. We have used this fact in many of our derivations. Now, consider two independent and normally distributed populations with unknown variances of and oz. If we get a random sample X1, X2, ..., Xn from the first population and a random sample Y1, Y2, ..., Yn from the second population (note that both samples are of equal size n), then can you recommend a hypothesis test to accept or reject that the two populations have the same mean? Explain what the setup of the hypothesis test is, which statistic you would use, and when you would accept or reject the null hypothesis.

Answer 1

Given Information: n.X and s n,, 7 and s, We should use t distribution, since the population standard deviation unknown. The hypothesis is given as, Hill. = H .Elly The null hypothesis states that there is not difference between two populations mean. The alternative hypothesis states that there is difference between two populations mean. The test statistics under null hypothesis is calculated as follows, X- (n,-1)s: +(, -1); - where sy n , tn, -2 Vnny We assume the population variance is equalfor the above test statistics. We reject null hypothesis when test statistics value is greater than critical value of t-distribution.