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

Step-by-step 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 : $\dfrac{\hat{p}-p}{\sqrt{\dfrac{p(1-p)}{n}}}$

$z=\dfrac{0.13-0.14}{\sqrt{\dfrac{0.14(1-0.14)}{478}}}\\\\=-0.630087269176\approx-0.63$

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