Question

In: Math

Let xi, i = 1,2,...,n be independent realizations from a population distributed like a Pareto with...

Let xi, i = 1,2,...,n be independent realizations from a population distributed like a Pareto with unknown mean.

(a) Compute the Standard Error of the sample mean. [Note: By“compute”, I mean “find the formula for”]. What is the probability that the sample mean is two standard deviations larger than the popula-
tion mean?

(b) Suppose the sample mean is 1 and the Standard Error is 2. Con- sider a test of the hypothesis that the population mean is 0, against the alternative that it is greater. What is the p−value? Compute a 5% confidence interval [Note: Use the Normal as an approx- imation for simplicity]. What is the type II error if the actual population mean is 2?

Solutions

Expert Solution

a) Let and be population mean and population standard deviation respectively.

Let = sample mean.

Standard error of sample mean is

Required Probability

By using Chebychev's inequality

( If X is a random variable with mean and Variance then for any positive integer k

)

Hence required probability is

Probability that sample mean is two standard deviation larger than population mean is atmost 0.25.

b ) From the information

We have to testing the hypothesis

against

The value of test statistic is

Under Ho the value of test statistic is

Since the test is right-tailed p-value is obtained by

from normal probability table

P ( Z > 0.50) = 0.3085

p-value = 0.3085.

5% confidence interval for population mean is

Alpha : level of significance = 0.95

From normal probability table

= (0.8746, 1.1254)

Since the test is right-tailed At 5% level of significance the critical region is Z > Zalpha/2

i.e.

Under Ho

Hence Critical region is and accepance region is

Probability of Type II error = P ( Accept Ho / Ha is true)

= P ( Z < 0.64)

From normal probability table

P ( Z < 0.64) = 0.7389 = P ( Type II error).

milcah answered 4 months ago
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