Consider the population {Alice. (F), Max M. Justin (M)}. Two people are randomly
ID: 3230975 • Letter: C
Question
Consider the population {Alice. (F), Max M. Justin (M)}. Two people are randomly selected with replacement (sample size is n = 2). (a) What is the proportion of men in the population? (b) List the set of ALL possible samples (with replacement, order always mattering). Calculate the sample proportion of each sample. (d) Calculate the mean of the sample proportions. Is it equal to the population proportion? (e) Give the sampling distribution of the sample proportion. Use a frequency table to represent your sampling distribution.Explanation / Answer
Solution:-
Given, Population - Alice (F), Max (M) and Justin (M).
Sample size, n = 2
(a) Proportion of men in the population = Total number of men / Total population
= 2 / 3
(b) Set of all possible samples,
As we need the sample of two, we choose any two people from the population randomly,
Alice and Max - 1/3 * 2/3 = 2/3
Alice and Justin - 1/3 *2/3 = 2/3
Max and Justin - 2/3 * 2/3 = 4/3
(d) Mean of sample proportions.
(2/3 + 2/3 +4/3) / 3 = 0.88
Mean of population proportion
(1/3 + 2/3 + 2/3) / 3 = 0.77
No, it is not equal. However, the mean of the sampling distribution (x) is equal to the mean of the population () when the population and sample size is large and distribution tends to normal distribution.
(e) We find that the mean of the sampling distribution of the proportion (p) is equal to the probability of success in the population (P). And the standard error of the sampling distribution (p) is determined by the standard deviation of the population (), the population size, and the sample size. These relationships are shown in the equations below:
If we make a frequency distribution table we will have 1 as frequecy for each data. So, the mean will be 1.
p = P
p = [ / sqrt(n) ] * sqrt[ (N - n ) / (N - 1) ]
p = sqrt[ PQ/n ] * sqrt[ (N - n ) / (N - 1) ]
where = sqrt[ PQ ].
Like the formula for the standard error of the mean, the formula for the standard error of the proportion uses the finite population correction, sqrt[ (N - n ) / (N - 1) ].
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