Question

Suppose 46% of American singers are Grammy award winners.

If a random sample of size 622 is selected, what is the probability that the proportion of Grammy award winners will be less than 47%? Round your answer to four decimal places.

Answer 1

The probability that the proportion of Grammy award winners will be less than 47% is 0.6915

Explanation:

* Lets explain how to solve the problem

- Suppose 46% of American singers are Grammy award winners.

- In a random sample of size 622 is selected

- We want to find the probability that the proportion of Grammy award

winners will be less than 47%

* At first we must to calculate z

∵
`z=\frac{P^(')-P}{\sqrt{(P(1-P))/(n)}}`, where

# P' is the sample proportion

# n is the sample size

# P is probability of success

∵ The sample proportion is 47% = 47/100 = 0.47

∴ P' = 0.47

∵ The sample size is 622

∴ n = 622

∵ The probability of success is 46% = 46/100 = 0.46

∴ P = 0.46

∴
`z=\frac{0.47-0.46}{\sqrt{(0.46(1-0.46))/(622)}}=0.5004`

- P(P' < 0.47) = P(z < 0.5004)

∵ P(z < 0.5004) = 0.6915

∴ P(P' < 0.47) = 0.6915

* The probability that the proportion of Grammy award winners will

be less than 47% is 0.6915