Question

Suppose that a random sample is to be taken from the Bernoulli distribution with an unknown parameter,p. Suppose also that it is believed that the value ofpis in the neighborhood of 0.2. How large must a random sample be taken so that\(P\left( {{\bf{|}}{{{\bf{\bar X}}}_{\bf{n}}}{\bf{ - p|}}} \right) \ge 0.75\)when p=0.2?

Short answer

The needed sample size is 10

Step-by-step solution

Given information

Referring to question Exercise 5

Finding sample size

When p=0.2, the random variable \({Z_n} = n{\bar X_n}\) will have a binomial distribution with parameters n and p=0.2, and

\(\Pr \left( {|{{\bar X}_n} - p| \le 0.1} \right) = \Pr \left( {0.1n \le {Z_n} \le 0.3n} \right)\)

The value of n for which this probability will be at least 0.75 must be determined by trial and error from the binomial distribution table at the back of the book. For n=8, the probability becomes

\(\begin{align}\Pr \left( {0.8 \le {Z_8} \le 2.4} \right) &= \Pr \left( {{Z_8} = 1} \right) + \Pr \left( {{Z_8} = 2} \right)\\ &= 0.3355 + 0.2639\\ &= 0.6291\end{align}\)

For n=9, they have

\(\begin{align}\Pr \left( {0.9 \le {Z_9} \le 2.7} \right) &= \Pr \left( {{Z_9} = 1} \right) + \Pr \left( {{Z_9} = 2} \right)\\ &= 0.3020 + 0.3020\\ &= 0.6040\end{align}\)

For n=10, they have

\(\begin{align}\Pr \left( {1 \le {Z_{10}} \le 3} \right) &= \Pr \left( {{Z_{10}} = 1} \right) + \Pr \left( {{Z_{10}} = 2} \right) + \Pr \left( {{Z_{10}} = 3} \right)\\ &= 0.2684 + 0.3020 + 0.2013\\ &= 0.7717\end{align}\)

Hence n=10 is sufficient.

It should be noted that although a sample size of n=10 will meet the required conditions, a sample size of n=11 will not meet the required conditions. For n=11, we would have

\(\Pr \left( {1.1 \le {Z_{11}} \le 3.3} \right) = \Pr \left( {{Z_{11}} = 2} \right) + \Pr \left( {{Z_{11}} = 3} \right)\)

Thus, only two terms of the binomial distribution for n=11 are included, whereas three of the binomial distribution for n=10 were included.