Question

8.92 Tony's Pizza guarantees all pizza deliveries within 30 minutes of the placement of orders. An agency wants to estimate the proportion of all pizzas delivered within 30 minutes by Tony's. What is the most conservative estimate of the sample size that would limit the margin of error to within .02 of the popula- tion proportion for a 99% confidence interval? 8.93 Refer to Exercise 8.92. Assume that a preliminary study has shown that 93% of all Tony's pizzas are delivered within 30 minutes. How large should the sample size be so that the 99% confidence interval for the population proportion has a margin of error of .02?

Answer 1

8.92)

Solution :

Given that,

$\hat p$ = 0.5

1 - $\hat p$ = 0.5

margin of error = E = 0.02

Z_$\alpha$/2 = 2.576

sample size = n = (Z_{$\alpha$ / 2} / E )² * $\hat p$ * (1 - $\hat p$ )

= (2.576 / 0.02)² * 0.5 * 0.5

= 4148

sample size = n = 4148

8.93

$\hat p$ = 0.93

1 - $\hat p$ = 0.07

margin of error = E = 0.02

Z_$\alpha$/2 = 2.576

sample size = n = (Z_{$\alpha$ / 2} / E )² * $\hat p$ * (1 - $\hat p$ )

= (2.576 / 0.02)² * 0.93 * 0.07

= 1080

sample size = n = 1080