位置8的问题 Random samples of size 100 are taken from a process (an infinite population) whose population proportion is 0.2. The mean and standard deviation of the distribution of sample proportions are _____. 20 and 0.040.2 and 0.040.2 and 0.2None of the alternative answers is correct.单项选择题
A
20 and 0.04
B
0.2 and 0.04
C
0.2 and 0.2
D
None of the alternative answers is correct.

Question

位置8的问题 Random samples of size 100 are taken from a process (an infinite population) whose population proportion is 0.2. The mean and standard deviation of the distribution of sample proportions are _____. 20 and 0.040.2 and 0.040.2 and 0.2None of the alternative answers is correct.单项选择题

A

20 and 0.04

B

0.2 and 0.04

C

0.2 and 0.2

D

None of the alternative answers is correct.

BlackTom AI · Accepted Answer

The question asks for the mean and standard deviation of the distribution of sample proportions when drawing samples of size 100 from an infinite population with population proportion p = 0.2.

First, recall that for a binomial-based sampling model, the sampling distribution of the sample proportion p̂ has a mean equal to the population proportion p. So the mean should be p = 0.2.

Second, the standard deviation (standard error) of p̂ is sqrt[p(1 − p) / n] for a sample size n. Here, p = 0.2 and n = 100, so the calculation is sqrt[0.2 × 0.8 / 100] = sqrt[0.16 / 100] = sqrt[0.0016] = 0.04.

Thus, the pair (mean, standard deviation) is (0.2, 0.04).

Now evaluate the options:
- Option "20 and 0.04": This is incorrect because the mean cannot be 20 for a proportion; proportions range between 0 and 1.
- Option "0.2 and 0.04": This matches the correct mean and standard deviation as derived above.
- Option "0.2 and 0.2": Here the standard deviation is miscomputed; 0.2 would imply a much larger variability than the actual 0.04 for n = 100, given p = 0.2.
- Option "None of the alternative answers is correct.": Since there is a correct option, this statement is false.

Additional note: Since the population is treated as infinite, there is no need to apply a finite population correction factor; the standard formula for the standard deviation of p̂ applies directly here.

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