Question

During the grand opening of the T-Shirt Shack at the Madera County Emu Farm, four of the emus broke through the fence and attacked the customers. The manage of the emu farm is now looking for data about insurance settlements. She has heard that the proportion of settlements that are favorable to businesses is 0.68. She wants to verify this number with her own study. She wants to be 99% confident in her results, and she wants the margin of error to be within 0.1. How many businesses does she need to sample to meet this criteria

Answer 1

Given :

p = 0.68

margin of error, E = 0.1

99% confident interval

alpha, α = 1 - 0.99

             = 0.01

$\frac{\alpha }{2}=\frac{0.01}{2} = 0.005$

∴ $z_{\alpha/2}=z_{0.005}=2.5758$ (z-critical value)

Sample size = $z_{\alpha/2} \times \frac{p(1-p)}{E^2}$

                    $=(2.5758)^2 \times 0.68 \times \frac{(1-0.68)}{0.1^2}$

                    $= 6.63475 \times 0.68 \times \frac{0.32}{0.01}$

                   = 144.37

                  = 145 (rounded off)

Therefore, there are 145 businesses does she need to sample to meet this criteria.