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Chapter 8: Q8 E (page 469)

For the conditions of Exercise 5, how large a random sample must be taken so that\({{\bf{E}}_{\bf{p}}}\left( {{\bf{|}}{{{\bf{\bar X}}}_{\bf{n}}} - {\bf{p}}{{\bf{|}}^{\bf{2}}}} \right) \le {\bf{0}}{\bf{.01}}\)for every possible value ofp (0鈮鈮�1)?

Short Answer

Expert verified

The needed sample size is \(n \ge 25\)

Step by step solution

01

Given information

Referring to question Exercise 5

02

Finding sample size

Here X is a Bernoulli random variable.

\(So,E\left( {{{\bar X}_n}} \right) = p\)

\(\begin{align}{E_p}\left( {|{{\bar X}_n} - p{|^2}} \right) &= Var\left( {{{\bar X}_n}} \right)\\ &= \frac{{p\left( {1 - p} \right)}}{n}\end{align}\)

This variance will be maximum when p=0.5

\(\begin{align}{E_p}\left( {|{{\bar X}_n} - p{|^2}} \right) &= Var\left( {{{\bar X}_n}} \right)\\ &= \frac{{0.5\left( {1 - 05} \right)}}{n}\\ &= \frac{{0.25}}{n}\end{align}\)

Here given that,

\(\begin{align}{E_p}\left( {|{{\bar X}_n} - p{|^2}} \right) \le 0.01\\\frac{{0.25}}{n} \le 0.01\\n \ge \frac{{0.25}}{{0.01}}\\n \ge 25\end{align}\)

So, the needed sample size is 25

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