Question

The director of the library calculates that 29% 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 412 books would differ from the population proportion by less than 4%? Round your answer to four decimal places.

Answer 1

Answer:

The probability that the proportion of books checked out in a sample of 412 books would differ from the population proportion by less than 4% is P=0.9255.

Step-by-step explanation:

The population's proportion is $\pi=0.29$

If we take a sample of 412 books (n=412), we have a standard deviation of the sampling distribution of:

$\sigma_M=\sqrt{\frac{\pi(1-\pi)}{n}}=\sqrt{\frac{0.29*0.71}{412}}=\sqrt{0.0005} =0.0224$

We have to calculate the probabilty that the sample proportion will be between 0.25 and 0.33.

$z_1=(0.25-0.29)/0.0224=-0.04/0.0224=-1.7857\\\\z_2=1.7857\\\\\\P(0.25<\hat{p}<0.33)=P(-1.7857<z<1.7857)\\\\P(0.25<\hat{p}<0.33)=P(z<1.7857)-P(z<-1.7857)\\\\P(0.25<\hat{p}<0.33)=0.9629-0.0371=0.9255$