Question

The opera theater manager believes that 15% of the opera tickets for tonight's show have been sold. If the manager is right, what is the probability that the proportion of tickets sold in a sample of 411411 tickets would be less than 10%

Answer 1

Answer:0.0023

Explanation:

Given: The proportion that the opera tickets for tonight's show have been sold : p =0.15

Sample size : n=411

Let
`\hat{p}` be the sample proportion.

The probability that the proportion of tickets sold in a sample of 411411 tickets would be less than 10% :

`P(\hat{p}<0.10)=P(\frac{\hat{p}-p}{\sqrt{(p(1-p))/(n)}}<\frac{0.10-0.15}{\sqrt{(0.15*(1-0.15))/(411)}})\\\\=P(Z<\frac{-0.05}{\sqrt{(0.1275))/(411)}})\ \ \ [Z=\frac{\hat{p}-p}{\sqrt{(p(1-p))/(n)}} ]\\\\=P(Z<-2.84)=1-P(Z<2.84)\\\\=1- 0.9977\\\\=0.0023`

Hence, the required probability = 0.0023