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Chapter 8: Q7 E (page 469)

For the conditions of Exercise 5, how large a random sample must be taken in order that\({{\bf{E}}_{\bf{p}}}\left( {{\bf{|}}{{{\bf{\bar X}}}_{\bf{n}}}{\bf{ - p}}{{\bf{|}}^{\bf{2}}}} \right) \le {\bf{0}}{\bf{.01}}\) whenp=0.2?

Short Answer

Expert verified

The needed sample size is \(n \ge 16\)

Step by step solution

01

Given information

Referring to question Exercise 5

02

Finding sample size

Here X is a Bernoulli random variable.

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

\(\begin{align}{E_p}\left( {|{{\bar X}_n} - p{|^2}} \right) &= Var\left( {{{\bar X}_n}} \right)\\ &= \frac{{p\left( {1 - p} \right)}}{n}\\ &= \frac{{0.2 \times 0.8}}{n}\,(For\,p = 0.2)\\ &= \frac{{0.16}}{n}\end{align}\)

Here given that,

\(\begin{align}{E_p}\left( {|{{\bar X}_n} - p{|^2}} \right) \le 0.01\\\ frac{{0.16}}{n} \le 0.01\\ n \ge \frac{{0.16}}{{0.01}}\\ n \ge 16\end{align}\)

So, the needed sample size is 16

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Most popular questions from this chapter

Suppose that\({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\)form a random sample from the exponential distribution with unknown parameter 尾. Show that if n is large, the distribution of the M.L.E. of 尾 will be approximately a normal distribution with mean 尾 and variance\(\frac{{{{\bf{\beta }}^{\bf{2}}}}}{{\bf{n}}}\).

Question:Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{, }}{\bf{. }}{\bf{. }}{\bf{. , }}{{\bf{X}}_{\bf{n}}}\) form n Bernoulli trials for which the parameter p is unknown (0鈮鈮�1). Show that the expectation of every function \({\bf{\delta }}\left( {{{\bf{X}}_{\bf{1}}}{\bf{, }}{\bf{. }}{\bf{. }}{\bf{. , }}{{\bf{X}}_{\bf{n}}}} \right)\)is a polynomial in p whose degree does not exceed n.

Sketch the p.d.f. of the\({{\bf{\chi }}^{\bf{2}}}\)distribution withmdegrees of freedom for each of the following values ofm. Locate the mean, the median, and the mode on each sketch. (a)m=1;(b)m=2; (c)m=3; (d)m=4.

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)?
  1. Construct a 2脳2 orthogonal matrix for which the first row is as follows: \(\left( {\begin{align}{}{\frac{{\bf{1}}}{{\sqrt {\bf{2}} }}}&{\frac{{\bf{1}}}{{\sqrt {\bf{2}} }}}\end{align}} \right)\)
  2. Construct a 3脳3 orthogonal matrix for which the first row is as follows: \(\left( {\begin{align}{}{\frac{{\bf{1}}}{{\sqrt {\bf{3}} }}}&{\frac{{\bf{1}}}{{\sqrt {\bf{3}} }}}&{\frac{{\bf{1}}}{{\sqrt {\bf{3}} }}}\end{align}} \right)\)
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