Suppose that 80% of the population likes cake. Say we sample 3 people from the p

Question

Suppose that 80% of the population likes cake.

Say we sample 3 people from the population, with replacement, and let X be the number of those 3 people who like cake. Find the pmf of X. (Recall this means finding pX(k), or in other words P(X = k), for all values of k.) Please calculate decimal approximations for your answers.

Suppose now that we sample 3 people without replacement from an overall population of size 10. (So the population consists of 8 people who like cake and 2 people who do not.) Again let X be the number of those 3 people who like cake, and find the pmf of X. As before, calculate decimal approximations.

Repeat the previous part with an overall population of size 1000. To get a decimal approximation, please use Mathematica, Wolfram Alpha, or similar software that can handle very large numbers. (The command for n k is binomial(n,k).)

Repeat the previous part with an overall population of size 1000000.

Compare your results, and comment on how population size affects the difference between sampling with and without replacement.

Explanation / Answer

Since 80% people like the cake, so p = 0.8

For sample size of 10:

With replacement:

P(X=0) = 3C0 * 0.80 * (1-0.8)3 = 0.008

P(X=1) = 3C1 * 0.81 * (1-0.8)2 = 0.096

P(X=2) = 3C2 * 0.82 * (1-0.8)1 = 0.384

P(X=3) = 3C3 * 0.83 * (1-0.8)0 = 0.512

Without replacement:

P(X=0) = 0 (because when we choose 3 out 10 people and only 2 people don't like the cake, there will always be a person who will like it)

P(X=1) = 8C1 * 2C2 / 10C3 = 0.067

P(X=2) = 8C2 * 2C1 / 10C3 = 0.467

P(X=3) = 8C3 * 2C0 / 10C3 = 0.467

For sample size of 1000:

Without replacement:

P(X=0) = 800C0 * 200C3 / 1000C3 = 0.0079

P(X=1) = 800C1 * 200C2 / 1000C3 = 0.0958

P(X=2) = 800C2 * 200C1 / 1000C3 = 0.3846

P(X=3) = 800C3 * 200C0 / 1000C3 = 0.5116

In the same way compute for the population size of 10^6 as well.

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