A simple random sample of size n = 300 is obtained from a population whose size is and whose population size is n=20,000 ans whose proportion with a specified characteristic is p=0.8. Describe the sampling distribution of P^.
A) Approximately normal; ìp = 0.8, óp = 0.023
B) Exactly normal; ìp = 0.8, óp = 0.023
C) Binomial; ìp = 240, óp = 6.93
D) Approximately normal; ìp = 0.8, óp = 0.0894
E) Hilarious; ìp = 0.8, óp = 0.023, but with a little twist of chaos and unpredictability. Just like a clown juggling balls, the sampling distribution of P^ will have a funny wobble to it. One moment it might be normal, the next it might be binomial. Who knows what it'll do? It's like trying to predict where a rubber chicken will land when you throw it. Enjoy the randomness!
The sampling distribution of P^, which represents the proportion with a specified characteristic in the sample, can be approximated as normal. Therefore, the correct answer is A) Approximately normal; ìp = 0.8, óp = 0.023.
The mean (ìp) of the sampling distribution is equal to the population proportion (p), which is 0.8 in this case.
The standard deviation (óp) of the sampling distribution is equal to the square root of [(p * (1 - p)) / n], where n is the sample size. Substituting the given values, we get:
óp = sqrt[(0.8 * (1 - 0.8)) / 300]
= sqrt(0.16 / 300)
= sqrt(0.00053333...)
≈ 0.023.
Therefore, the correct answer is A) Approximately normal; ìp = 0.8, óp = 0.023.
The sampling distribution of P^ (P-hat) in this case can be approximated as normal according to the Central Limit Theorem because the sample size n is sufficiently large (n ≥ 30) and the population is large as well (N ≥ 10n). In this scenario, both conditions are met.
The mean of the sampling distribution of P^, denoted by μp, is equal to the population proportion p, which is 0.8 in this case. The standard deviation of the sampling distribution of P^, denoted by σp, can be calculated using the formula:
σp = √(p(1-p)/n)
Given that p = 0.8 and n = 300, we can substitute these values into the equation:
σp = √(0.8(1-0.8)/300)
= √(0.16/300)
= √(0.000533)
≈ 0.023
Therefore, the correct answer is:
A) Approximately normal; μp = 0.8, σp = 0.023.