Question

Suppose an experimenter wishes to test

H0: µ = 100 vs H1: µ ≠ 100

at the α = 0.05 level of significance and wants 1 – β to equal 0.60 when µ = 103. What is the smallest (i.e. cheapest) sample size that will achieve the objective? Assume the variable being measured is normally distributed with σ = 14.

Answer 1

Solution:

Given:

H0: µ = 100 vs H1: µ ≠ 100

Level of Significance = α = 0.05

Power of the test = 1 – β = 0.60

µ = 103

the variable being measured is normally distributed with σ = 14 .

We have to find smallest (i.e. cheapest) sample size that will achieve the objective.

Formula:

where

α = 0.05 then α/2 = 0.05/2=0.025

thus 1 - α/2 = 1 - 0.025 = 0.9750

Look in z table for Area = 0.9750 or its closest area and find z value.

Area = 0.9750 corresponds to 1.9 and 0.06 , thus z critical value = 1.96

That is :

1 – β = 0.60

then

Look in z table for Area = 0.6000 or its closest area and find z critical value:

Area 0.5987 closest to 0.6000

and it corresponds to 0.2 and 0.05

thus z = 0.25

That is:

Thus