.2 3. Let X,, X, be a random sample from the N(0, 0) distribution, and let...

 

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.2 3. Let X,, X, be a random sample from the N(0, 0) distribution, and let Y,..., Ym be a random sample from the N(1, 0) distribution that is independent of the Xi's. Here, and are unknown. Define = 0}}/0}. .2 Y = (a) Find a size a test for H : o versus H : \ % \ 0 based on the statistic 1X/; X(Y - 1). Give the rejection region in terms of quantiles of the F distribution. m j=1 k i=1 2 (Hint: You can use the fact that if Z, ..., Zk are independent standard normal random variables, then - Z follows the chi-square distribution with k degrees of freedom. Also, if U is a chi-square random variable with pu degrees of freedom, V is a chi-square U/Pu random variable with py degrees of freedom, and U and V are independent, then V/pv follows the F distribution with degrees of freedom (pu, pv).) (b) Based on the test in (a), find a (1 - a) confidence interval for . .2 3. Let X,, X, be a random sample from the N(0, 0) distribution, and let Y,..., Ym be a random sample from the N(1, 0) distribution that is independent of the Xi's. Here, and are unknown. Define = 0}}/0}. .2 Y = (a) Find a size a test for H : o versus H : \ % \ 0 based on the statistic 1X/; X(Y - 1). Give the rejection region in terms of quantiles of the F distribution. m j=1 k i=1 2 (Hint: You can use the fact that if Z, ..., Zk are independent standard normal random variables, then - Z follows the chi-square distribution with k degrees of freedom. Also, if U is a chi-square random variable with pu degrees of freedom, V is a chi-square U/Pu random variable with py degrees of freedom, and U and V are independent, then V/pv follows the F distribution with degrees of freedom (pu, pv).) (b) Based on the test in (a), find a (1 - a) confidence interval for .