Suppose x equals the number of heads observed when a single

Suppose x equals the number of heads observed when a single coin is tossed; that is, x=0 or x=1. The population corresponding to x is the set of 0\'s and 1\'s generated when the coin is tossed repeatedly a large number of times. Suppose we select n=2 observations from this population. (That is, we toss the coin twice and observe two values of x).

a) List the three different samples (combinations of 0\'s and 1\'s) that could be obtained.

b) Calculate the value of x-bar for each of the samples.

c) List the values that x-bar can assume, and find the probabilities of observing these values.

Solution

(a) Three different samples are: {x=0,x=0}, {x=0,x=1},{x=1,x=1}

(b) Value of x bar(i.e. mean) = 0, 0.5,1

(c) x bar can assume 0, 0.5 or 1

Total possible events and their mean are

{x=0,x=0}, with mean 0

{x=0,x=1},with mean 0.5

{x=1,x=0} with mean 0.5

{x=1,x=1} with mean 1

Hence, probability of having 0 as mean = 1/4

probability of having 0.5 as mean = 2/4=1/2

probability of having 1 as mean = 1/4