Question

15. Sheldon Cooper is fluent in Klingon (including knowing how to say “Revenge is a dish best served cold” to Wil Wheaton). It got him to think about how many people are fluent in the language. If Sheldon would like to construct a 95% confidence interval for the proportion of people that are fluent, and would like his interval to be no more than 10 percentage points in width, how large a sample should he take?

Answer 1

Solution :

Given that,

$\hat p$ $\hatp$ = 0.5

1 - $\hat p$ = 1 - 0.5 = 0.5

width = 10% = 0.10

margin of error = E = width / 2 = 0.10 / 2 = 0.05

At 95% confidence level the z is ,

$\alpha$ = 1 - 95% = 1 - 0.95 = 0.05

$\alpha$ / 2 = 0.05 / 2 = 0.025

Z _{$\alpha$ /2} = Z_0.025 = 1.96

Sample size = n = ((Z _{$\alpha$ / 2}) / E)² * $\hat p$ * (1 - $\hat p$ )

= (1.96 / 0.05)² * 0.5 * 0.5

= 384.16

= 385

n = sample size = 385

385 sample should he take .