Question

Chapter 6, Section 2-CI, Exercise 109

What Influences the Sample Size Needed?

In this exercise, we examine the effect of the margin of error on determining the sample size needed. Find the sample size needed to give, with 95% confidence, a margin of error within ±, 8 . Within. ±, 5 Within ±, 1. Assume that we use ά= 25 as our estimate of the standard deviation in each case. Round your answers up to the nearest integer.

ME= 8: n= _______

ME=5: n= _______

ME= 1: n= _______

Answer 1

Solution :

Given that,

standard deviation = =25

a ) margin of error = E =8

At 95% confidence level the z is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z/2 = Z0.025 = 1.960

Sample size = n = ((Z/2 * ) / E)2

= ((2.576 * 25) / 8)2

= 38

Sample size = 38

b ) margin of error = E =5

At 95% confidence level the z is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z/2 = Z0.025 = 1.960

Sample size = n = ((Z/2 * ) / E)2

= ((2.576 * 25) / 5)2

= 96

Sample size = 96

a ) margin of error = E =1

At 95% confidence level the z is ,

= 1 - 95% = 1 - 0.95 = 0.05

/ 2 = 0.05 / 2 = 0.025

Z/2 = Z0.025 = 1.960

Sample size = n = ((Z/2 * ) / E)2

= ((2.576 * 25) / 1)2

= 2401

Sample size = 2401