A sampling distribution (e.g. of the sample mean) is a distribution, not a histo

Question

A sampling distribution (e.g. of the sample mean) is a distribution, not a histogram of observed sample meansi the histogram of sample means discussea ln class 1s Just an lntuitive way o thinking about the sampling distribution; technically, it's called the *empirical* sampling dist bution. Of course, if the number of trials is infinite, then the empirical sampling distribution (i.e., the histogram) approaches the distribution. Anyway, to show that the sampling distribution is truly a distribution (not a histogram), let's derive one mathematically -no data at all. Consider a population described by a Bernoulli random variable, i.e., x = 0,1, following the Bernau 111 distribution, i.e., p(x) = p11x (1-p1)^ (1-x) . Suppose we take samples of size 2. a) Write down all the possible samples. Hint: there are only 4 b) For each of the possible samples, compute the sample mean. each of the possible samples, compute the probability. Hint: Use Bernoulli d) Based on your answers to parts a-c, find the probability of each of the possible sample meanS. Note your answer to part d *is* the sampling distribution of the sample mean! Note that it's not a histogram, but a real distribution.

Explanation / Answer

a) All the four possible samples are

Sample 1: (0,0)

Sample 2: (0,1)

Sample 3: (1,0)

Sample 4: (1,1).

b) Sample means are:

Sample 1: Mean=(0+0)/2=0

Sample 2: Mean=(0+1)/2=0.5

Sample 3: Mean=(1+0)/2=0.5

Sample 4: Mean=(1+1)/2=1

c) The probability of each sample is:

Sample 1: Probability = P(0)*P(0) = (1-p)*(1-p) = (1-p)2

Sample 2: Probability = P(0)*P(1) = (1-p)*p = p(1-p)

Sample 3: Probability = P(1)*P(0) = p*(1-p) = p(1-p)

Sample 4: Probability = P(1)*P(1) = p*p = p2

d) Sample mean                                  Probability

        0                                                     (1-p)2

       0.5                                                  2p(1-p)

        1                                                      p2

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