Question

FF1:18 1H20B B 80 ma2500a16-1 ma2500s14 ma2500a15 ma2500s15 ma2500a17 2. Let Xi, X2 , X10 be a random sample of observations from the N(μ, σ*) distribution where μ is unknown and σ2-10. We reject the null hypothesis Ho : μ-5 in lavour of the alternative hypothesis H1 : μ < 5 if sum of the observations is less than or equal to 35 (a) What is the critical region for the test? (b) Compute the size of the test (c) Compute the power of the test at 11-1 and μ-3 13] 151 17] Let (2, IP) be a probability space and let X be a random variable on 2. (a) What is the distribution of X? (b) Show that the distribution of X is a probability measure on subsets of R. (c) Let X ~ Exponential (A) where λ > 0 is a rate parameter. Find the CDF 3. of the random variable Yce where cis a constant 4. Let X be continuous random variable whose PDF is given by f(x:0) = otherwise where 6 > 0 is an unknown scalar parmeter. (a) Find the maximum likelihood estimator of θ. (b) Find a method-of-moments estimator of 0 for the case when 151 151 1SI > 1 n) Whv n wß not find mathíd-nf.ma mP nte t 1 mator whρ n A <. 1 ?

Answer 1

3.

(a)

We reject the null hypothesis is the sum of observations $\le$ 35. That is we reject the null hypothesis is the sample mean $\le$ 35/10 = 3.5

The critical region of the test is sample mean = 3.5. That is, if sample mean $\le$ 3.5, we reject H0, else we accept H0.

(b)

Standard deviation of sample mean = $\sigma/\sqrt{n} = \sqrt{10} / \sqrt{10}$ = 1

Size of the test = P(falsely rejecting the null hypothesis) = P(sample mean $\le$ 3.5 | $\mu$ = 5)

= P[Z $\le$ (3.5 - 5)/ 1]

= P[Z $\le$ -1.5]

= 0.0668

(c)

Power of the test at $\mu$ = 1 = P(Reject H0 | $\mu$ = 1) = P(sample mean $\le$ 3.5 | $\mu$ = 1)

= P[Z $\le$ (3.5 - 1)/ 1]

= P[Z $\le$ 2.5]

= 0.9938

Power of the test at $\mu$ = 3 = P(Reject H0 | $\mu$ = 3) = P(sample mean $\le$ 3.5 | $\mu$ = 3)

= P[Z $\le$ (3.5 - 3)/ 1]

= P[Z $\le$ 0.5]

= 0.6915