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

Question

By R.

A sampling distribution (e.g. of the sample mean) is a distribution, not a histogram of observed sample means: the histogram of sample means discussed in class is just an intuitive way of thinking about the sampling distribution: technically, it's called the *empirical* sampling distribution. 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 Bernoulli distribution, i e., p(x) = pi^x (1 - pi)^(1 - x). Suppose we take samples of size 2. a) Write down all the possible samples. b) For each of the possible samples, compute the sample mean c) For each of the possible samples, compute the probability. 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 possible samples is

(0,0), (0,1), (1,0), (1,1)

b) For each possible samples, the sample mean is 0,0.5,0.5,1

c) P(X=0, X=0)=(1-p)2

P(X=1, X=0)=p(1-p)

P(X=0, X=1)=p(1-p)

P(X=1,X=1) = p2

d) The probability of each of the possible sample means are

P(X=0)= (1-p)

P(X=0.5)= p0.5(1-p)0.5

P(X=0.5)= p0.5(1-p)0.5

P(X=1)= p

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