Answer:
The probability that the proportion of Grammy award winners will be less than 47% is 0.6915
Step-by-step 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
∵ [tex]z=\frac{P^{'}-P}{\sqrt{\frac{P(1-P)}{n}}}[/tex], 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
∴ [tex]z=\frac{0.47-0.46}{\sqrt{\frac{0.46(1-0.46)}{622}}}=0.5004[/tex]
- 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
