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 distribution of p^ (choose the correct phrase that best describes the shape of the sampling below)
a) Not normal because n <, 0.05N and np (1-p) <10.
b) Approximately normal because n <, 0.05N and np (1-p) >10.
c) Approximately 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^ = ______ (round six decimal places as needed)
4) What is the probability of obtaining x=123 or more individuals with the characteristic? 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)
(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 is less than 0.05 times the population size N=20,000, and np(1-p) = 150 * 0.8 * (1-0.8) = 24, which is greater than 10. These conditions indicate that the sampling distribution of p^ (sample proportion) is approximately normal.
(2) The mean of the sampling distribution of p^ is equal to the population proportion p, which is 0.8. Therefore, the mean is p^ = 0.8.
(3) The standard deviation of the sampling distribution of p^ is calculated using the formula:
Standard Deviation of p^ = sqrt((p(1-p))/n)
Substituting the values, we get:
Standard Deviation of p^ = sqrt((0.8 * (1-0.8))/150)
Calculating this, we get:
Standard Deviation of p^ = 0.0259 (rounded to six decimal places)
(4) To find the probability of obtaining p^ > 0.82, we need to calculate the z-score and then find the corresponding probability using a standard normal distribution table.
The z-score can be calculated using the formula:
z = (p^ - p) / sqrt((p(1-p))/n)
Substituting the values, we get:
z = (0.82 - 0.8) / sqrt((0.8 * (1-0.8))/150)
Calculating this, we get:
z = 0.08 / 0.0259
z = 3.083
Using the z-score table or a calculator, we find that the probability corresponding to a z-score of 3.083 is approximately 0.9992.
Therefore, P(p^ > 0.82) = 1 - P(p^ ≤ 0.82) = 1 - 0.9992 = 0.0008 (rounded to four decimal places)
(5) To find the probability of obtaining p^ < 0.74, we can follow similar steps as in question 4.
Calculating the z-score:
z = (0.74 - 0.8) / sqrt((0.8 * (1-0.8))/150)
z = -0.06 / 0.0259
z = -2.316
Using the z-score table or a calculator, we find that the probability corresponding to a z-score of -2.316 is approximately 0.0107.
Therefore, P(p^ < 0.74) = 0.0107 (rounded to four decimal places)