Question

The director of the library believes that 13% of the library's collection is checked out. If the director is right, what is the probability that the proportion of books checked out in a sample of 774 books would be less than 10%

Answer 1

The probability that the proportion of books checked out in a sample of 774 books would be less than 10% is 0.0066

Explanation:

We are given that The director of the library believes that 13% of the library's collection is checked out.

`\mu = p = 0.13`

Total no. of books = 774

Standard deviation =
`\sqrt{(p(1-p))/(n)}`

Standard deviation =
`\sqrt{(0.13(1-0.13))/(774)}=0.012`

We are supposed to find the probability that the proportion of books checked out in a sample of 774 books would be less than 10%

`P(\hat{p}<0.1)\\Z=(x-\mu)/(\sigma)\\Z=(0.1-0.13)/(0.012)\\Z=-2.48\\P(Z<-2.48)=0.0066`

So,
`P(\hat{p}<0.1)=0.0066`

Hence the probability that the proportion of books checked out in a sample of 774 books would be less than 10% is 0.0066