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Chapter 7: Q.AP2.25 - Cumulative AP Practise Test (page 438)

In a city library, the mean number of pages in a novel is 525 with a standard deviation of 200. Approximately 30%of the novels have fewer than 400 pages. Suppose that you randomly select 50 novels from the library.

a. What is the probability that the average number of pages in the sample is less than 500 ?

b. What is the probability that at least 20 of the novels have fewer than 400 pages?

Short Answer

Expert verified

(a)the probability that the total number of pages is less than 25,000 is 0.1894

(b)Then approximation the binomial distribution by the normal distribution渭p^=p=0.30

Step by step solution

01

Part (a) Step 1: Given Information

渭=525蟽=200n=50

Formula used:

z=x炉-渭蟽/n

02

Part (a) Step 2: Simplification

Sample mean is

25,0050=500z=x炉-渭蟽/n=500-525200/50=-0.88P(x炉<500)=P(Z<-0.88)=0.1894

Therefore the probability that the total number of pages is less than 25,000 is 0.1894

03

Part (b) Step 1: Given Information

n=50p=30%=0.30

Formula used:

蟽p^=p(1-p)nz=x-渭蟽

04

Part (b) Step 2: Simplification

For a normal approximation of the binomial distribution: np鈮�and nq鈮�10.

np=50(0.30)=15鈮�10nq=n(1-p)=50(1-0.35)=35鈮�10

Then approximation the binomial distribution by the normal distribution

渭p^=p=0.30

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