Answer:0.0023
Step-by-step explanation:
Given: The proportion that the opera tickets for tonight's show have been sold : p =0.15
Sample size : n=411
Let [tex]\hat{p}[/tex] be the sample proportion.
The probability that the proportion of tickets sold in a sample of 411411 tickets would be less than 10% :
[tex]P(\hat{p}<0.10)=P(\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}<\dfrac{0.10-0.15}{\sqrt{\dfrac{0.15\times(1-0.15)}{411}}})\\\\=P(Z<\dfrac{-0.05}{\sqrt{\dfrac{0.1275)}{411}}})\ \ \ [Z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}} ]\\\\=P(Z<-2.84)=1-P(Z<2.84)\\\\=1- 0.9977\\\\=0.0023[/tex]
Hence, the required probability = 0.0023
