Suppose a simple random sample of size n=150 is obtained from a population whose size is N=20,000 and whose population proportion with a specified characteristic is p=0.8. Please answer questions (1) through (5) below.
(1) describe the sampling ditriribution of p^ (choose the corrct phrase that best decribes the shape of the sampling below)
a) Not normal because n <, 0.05N and np (1-p) <10.
b) Approximateley normal because n <, 0.05N and np (1-p) >10.
c) Approximateley normal because n <, 0.05N and np (1-p) <10.
d) Not normal because n <, 0.05N and np (1-p) >10.
2) Determine the mean of the sampling distribution of p^.
p^ = ______ (round to one decimal place as needed)
3) Determine the standard deviation of the sampling distribution of p^.
p^ = ______ (roaund six decimal places as needed)
4) What is the probability of obtaining x=123 or more individuals with the characterstic? that is , what P(p^ > 0.82)?
P( p^ > 0.82) = _____ (Round four decimals places)
5) What is the probability of obtaining x = 111 of fewer individuals with the characteristic ? that is what is P(p^ < 0.74)?
P(p^ < 0.74) = _______ (Round to four decimal places as needed)
Same as your other post:
http://www.jiskha.com/display.cgi?id=1417003909
(1) The correct phrase that best describes the shape of the sampling distribution is b) Approximately normal because n < 0.05N and np(1-p) > 10. In this case, n=150, N=20,000, and p=0.8. Since n is less than 0.05 times N and np(1-p) is greater than 10, the sampling distribution of p^ is approximately normal.
(2) To determine the mean of the sampling distribution of p^, we use the formula:
Mean of p^ = p = 0.8
So, the mean of the sampling distribution of p^ is 0.8.
(3) To determine the standard deviation of the sampling distribution of p^, we use the formula:
Standard deviation of p^ = √(p*(1-p)/n)
Standard deviation of p^ = √(0.8*(1-0.8)/150)
Standard deviation of p^ ≈ 0.027 (rounded to six decimal places)
So, the standard deviation of the sampling distribution of p^ is approximately 0.027.
(4) To find the probability of obtaining x=123 or more individuals with the characteristic, we need to find P(p^ > 0.82). We can use the normal distribution to approximate this probability.
First, we standardize the value of p^:
Z = (0.82 - p) / √(p*(1-p)/n)
Z = (0.82 - 0.8) / √(0.8*(1-0.8)/150)
Z ≈ 0.54
Next, we look up the probability corresponding to this Z-value in the standard normal distribution table or use a calculator. The probability is approximately 0.2946 (rounded to four decimal places).
So, P(p^ > 0.82) ≈ 0.2946.
(5) Similarly, to find the probability of obtaining x=111 or fewer individuals with the characteristic, we need to find P(p^ < 0.74). We again use the normal distribution to approximate this probability.
First, we standardize the value of p^:
Z = (0.74 - p) / √(p*(1-p)/n)
Z = (0.74 - 0.8) / √(0.8*(1-0.8)/150)
Z ≈ -1.63
Next, we look up the probability corresponding to this Z-value in the standard normal distribution table or use a calculator. The probability is approximately 0.0526 (rounded to four decimal places).
So, P(p^ < 0.74) ≈ 0.0526.