Question

The director of the library believes that 14% 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 478 books would be less than 13%? Round your answer to four decimal places.

Answer 1

Answer: 0.2643

Explanation:

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

As per given question , we have

`p=0.14\\\\ \hat{p}=0.13\\\\ n=478`

z-score :
`\frac{\hat{p}-p}{\sqrt{(p(1-p))/(n)}}`

`z=\frac{0.13-0.14}{\sqrt{(0.14(1-0.14))/(478)}}\\\\=-0.630087269176\approx-0.63`

The required probability ( using z-table ):-

`P(z<-0.63)=1-P(z<0.63)\\\\=1- 0.7356527=0.2643473\approx0.2643`

Hence, the probability that the proportion of books checked out in a sample of 478 books would be less than 13% = 0.2643