Suppose that independent random samples of sizenare taken from two normal popula

Question

Suppose that independent random samples of sizenare taken from two normal populations with known variances 21and 22 are used to test the null hypothesis 12=0 against the alternative hypothesis 12=a. Suppose further that the probabilities of type I and type II errors are to have values and respectively.

Problem 4 Suppose that independent random samples of size n are taken from two normal populations with known variances | and 1 are used to test the null hypothesis 1-12-dg against the alternative hypothesis 1-12 = 5a. Suppose further that the probabilities of type I and type II errors are to have values and respectively. (a) Show that the required sample size is given by n= 06-5a)2 (b) Find the required sample size if ,-9, 2-13, _ 80, a-86, 0.01, and -0.01.

Explanation / Answer

Here as type I error is .

H0 : 1 - 2 = 0

Ha : 1 - 2 = a  

Lets say required sample size = n

so first we have to find the sample mean difference xd  , above which we shall be able to reject the null hypothesis.

xd = 0 + Z sqrt (12/n)

Here, if type II error is so that means we cannot reject the null hypothesis even if it is true, that means if sample mean difference woulde be less than xd , then we cannot reject the null hypothesis.

so,

xd = a   - Z (22/n)

so comparing both values.

0 + Z sqrt (12/n) = a   - Z (22/n)

n = (12 + 22( (Z + Z)2/ (0 - a)2

(b) Here 1  = 9 ; 2  = 13 ; 0  = 80 ; a   = 86

Z = Z0.005 = 2.575 ;  Z   = Z0.01 = 2.326

n = (12 + 22) (Z + Z)2/ (0 - a)2

n = (92 + 132) * (2.575 + 2.326)2 /(86 - 80)2

n = 250 * 24.02 / 36 = 166.81

n = 167

so n could be 167.

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