Answer to Question #149406 in Statistics and Probability for jee

Question

a. The probability of success on any trial of a binomial
experiment is 25%. Find the probability
that the proportion of successes in a sample of
500 is less than 22%.
b. Repeat part (a) with n = 800.
c. Repeat part (a) with n = 1,000.

Answer 1

a. Find the probability of success that the proportion of success in a sample of 500 is less than 22% .

The probability of success for any trail of a binomial experiment is, p=0.25.

The sample size, n = 500

The probability of success that the proportion of success in a sample of 500 is less than 22% is,

$P(\bar{P}<0.22)=P( \frac{\bar{P}-p}{\sqrt{\frac{p(1-p)}{n}}}<\frac{0.22-0.25}{\sqrt{\frac{0.25(1-0.25)}{500}}} ) \\ =P(Z<\frac{0.22-0.25}{0.0194} \\ =p(Z<-1.55) \\ = 0.0606$

Therefore, the probability of success that the proportion of success in a sample of 500 is less than 22% is 0.0606.

b. Find the probability of success that the proportion of success in a sample of 800 is less than 22% .

The probability of success for any trail of a binomial experiment is, p=0.25.

The sample size, n=800

The probability of success that the proportion of success in a sample of 800 is less than 22% is,

$P(\bar{P}<0.22)=P( \frac{\bar{P}-p}{\sqrt{\frac{p(1-p)}{n}}}<\frac{0.22-0.25}{\sqrt{\frac{0.25(1-0.25)}{800}}} ) \\ =P(Z<\frac{0.22-0.25}{0.0153} \\ =p(Z<-1.96) \\ = 0.0250$

Therefore, the probability of success that the proportion of success in a sample of 800 is less than 22% is 0.0250.

c. Find the probability of success that the proportion of success in a sample of 1000 is less than 22%.

The probability of success for any trail of a binomial experiment is, p=0.25.

The sample size, n=1000

The probability of success that the proportion of success in a sample of 1000 is less than 22% is,

$P(\bar{P}<0.22)=P( \frac{\bar{P}-p}{\sqrt{\frac{p(1-p)}{n}}}<\frac{0.22-0.25}{\sqrt{\frac{0.25(1-0.25)}{1000}}} ) \\ =P(Z<\frac{0.22-0.25}{0.0137} \\ =p(Z<-2.19) \\ = 0.0143$

Therefore, the probability of success that the proportion of success in a sample of 1000 is less than 22% is 0.0143.