Question

Chapter 6, Section 2-CI, Exercise 111

What Influences the Sample Size Needed?

In this exercise, we examine the effect of the value of the estimated standard deviation on determining the sample size needed.

Find the sample size needed to give, with 95% confidence, a margin of error within ±5 , if the estimated standard deviation is ά= 50. If the estimated standard deviation is ά=20. If the estimated standard deviation is ά=10. Round your answers up to the nearest integer.

ά= 50: n=________

ά= 20: n= _________

ά= 10: n=_______

Answer 1

Solution :

Given that,

a ) standard deviation = =50

  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

= ((1.960 *50) / 5)2

=384

Sample size = 384

b ) standard deviation = =20

  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

= ((1.960 *20) / 5)2

= 61

Sample size =61

c ) standard deviation = =10

  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

= ((1.960 *10) / 5)2

= 15

Sample size = 15