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Chapter 8: Q 4E (page 494)

Suppose that\({X_1},...,{X_n}\)form a random sample from the normal distribution with unknown mean 渭 and known variance\({\sigma ^2}\). How large a random sample must be taken in order that there will be a confidence interval for 渭 with confidence coefficient 0.95 and length less than 0.01蟽?

Short Answer

Expert verified

153665

Step by step solution

01

Given information

\({X_1},...,{X_n}\) form a random sample from the normal distribution with unknown mean 渭 and known variance \({\sigma ^2}\).We need to calculate the minimum sample size that must be taken in order that there will be a confidence interval for 渭 with confidence coefficient 0.95 and length less than 0.01蟽.

02

Calculation of the sample size

Limits for confidence coefficient \(\gamma \) in the interval \(\left( {A,B} \right)\) is given by

\(\begin{align}A &= \bar X - {\Phi ^{ - 1}}\left( {\frac{{1 + \gamma }}{2}} \right)\frac{\sigma }{{\sqrt n }}\\B &= \bar X + {\Phi ^{ - 1}}\left( {\frac{{1 + \gamma }}{2}} \right)\frac{\sigma }{{\sqrt n }}\end{align}\)

Where \(\gamma \in \left( {0,1} \right)\)

For \(\gamma \)=0.95 we have

\(\begin{align}{\Phi ^{ - 1}}\left( {\frac{{1 + 0.95}}{2}} \right)\\ &= {\Phi ^{ - 1}}\left( {0.975} \right)\\ &= 1.96\end{align}\)

The length of the confidence interval is

\(\begin{align}L &= B - A\\ &= 2{\Phi ^{ - 1}}\left( {\frac{{1 + \gamma }}{2}} \right)\frac{\sigma }{{\sqrt n }}\\ &= 2 \times 1.96\frac{\sigma }{{\sqrt n }}\\ &= 3.92\frac{\sigma }{{\sqrt n }}\end{align}\)

According to the question

\(\begin{align}3.92\frac{\sigma }{{\sqrt n }} < 0.01\sigma \\n > 153665\end{align}\)

Hence the minimum sample size is 153665

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Most popular questions from this chapter

In the situation of Exercise 9, suppose that a prior distribution is used for胃with p.d.f.尉(胃)=0.1 exp(鈭�0.1胃)for胃 >0. (This is the exponential distribution with parameter 0.1.)

  1. Prove that the posterior p.d.f. of胃given the data observed in Exercise 9 is

\({\bf{\xi }}\left( {{\bf{\theta }}\left| {\bf{x}} \right.} \right){\bf{ = }}\left\{ \begin{align}{l}{\bf{4}}{\bf{.122exp}}\left( {{\bf{ - 0}}{\bf{.1\theta }}} \right)\;{\bf{if}}\;{\bf{4}}{\bf{.8 < \theta < 5}}{\bf{.2}}\\{\bf{0}}\;{\bf{otherwise}}\end{align} \right.\)

  1. Calculate the posterior probability \(\left| {{\bf{\theta - }}{{{\bf{\bar X}}}_{\bf{2}}}} \right|{\bf{ < 0}}{\bf{.1}}\), which \({{\bf{\bar X}}_{\bf{2}}}\)is the observed average of the datavalues.
  2. Calculate the posterior probability that胃is in the confidence interval found in part (a) of Exercise 9.
  3. Can you explain why the answer to part (b) is so close to the answer to part (e) of Exercise 9? Hint:Compare the posterior p.d.f. in part (a) to the function in Eq. (8.5.15).

Suppose that the five random variables \({{\bf{X}}_{\bf{1}}}{\bf{, \ldots ,}}{{\bf{X}}_{\bf{5}}}\) are i.i.d. and that each has the standard normal distribution. Determine a constantcsuch that the random variable

\(\frac{{{\bf{c}}\left( {{{\bf{X}}_{\bf{1}}}{\bf{ + }}{{\bf{X}}_{\bf{2}}}} \right)}}{{{{\left( {{\bf{X}}_{\bf{3}}^{\bf{2}}{\bf{ + X}}_{\bf{4}}^{\bf{2}}{\bf{ + X}}_{\bf{5}}^{\bf{2}}} \right)}^{\frac{{\bf{1}}}{{\bf{2}}}}}}}\)

will have atdistribution.

Question:Suppose that a random variable X has a normal distribution for which the mean 渭 is unknown (鈭掆垶 <渭< 鈭�) and the variance 蟽2 is known. Let\({\bf{f}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)\)denote the p.d.f. of X, and let\({\bf{f'}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)\)and\({\bf{f''}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)\)denote the first and second partial derivatives with respect to 渭. Show that

\(\int_{{\bf{ - }}\infty }^\infty {{\bf{f'}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)} {\bf{dx = 0}}\,\,{\bf{and}}\,\,\int_{{\bf{ - }}\infty }^\infty {{\bf{f''}}\left( {{\bf{x}}\left| {\bf{\mu }} \right.} \right)} {\bf{dx = 0}}{\bf{.}}\).

Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{, }}{\bf{. }}{\bf{. }}{\bf{. , }}{{\bf{X}}_{\bf{n}}}\) form a random sample from the normal distribution with unknown mean \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\), and also that the joint prior distribution of \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\) is the normal-gamma distribution satisfying the following conditions: \({\bf{E}}\left( {\bf{\mu }} \right){\bf{ = 0}}\,\,\,\,{\bf{,E}}\left( {\bf{\tau }} \right){\bf{ = 2,E}}\left( {{{\bf{\tau }}^{\bf{2}}}} \right){\bf{ = 5}}\,\,\,{\bf{and}}\,\,{\bf{Pr}}\left( {\left| {\bf{\mu }} \right|{\bf{ < 1}}{\bf{.412}}} \right){\bf{ = 0}}{\bf{.5}}\)Determine the prior hyperparameters \({{\bf{\mu }}_{\bf{0}}}{\bf{,}}{{\bf{\lambda }}_{\bf{0}}}{\bf{,}}{{\bf{\alpha }}_{\bf{0}}}{\bf{,}}{{\bf{\beta }}_{\bf{0}}}\)

Suppose that \({{\bf{X}}_{\bf{1}}}{\bf{, }}{\bf{. }}{\bf{. }}{\bf{. , }}{{\bf{X}}_{\bf{n}}}\) form a random sample from the normal distribution with unknown mean \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\), and also that the joint prior distribution of \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\) is the normal-gamma distribution satisfying the following conditions: \({\bf{E}}\left( {\bf{\tau }} \right){\bf{ = 1,Var}}\left( {\bf{\tau }} \right){\bf{ = }}\frac{{\bf{1}}}{{\bf{3}}}\,\,\,{\bf{and}}\,\,{\bf{Pr}}\left( {{\bf{\mu > 3}}} \right){\bf{ = 0}}{\bf{.5}}\,\,{\bf{and}}\,\,{\bf{Pr}}\left( {{\bf{\mu > 0}}{\bf{.12}}} \right){\bf{ = 0}}{\bf{.9}}\,\)

Determine the prior hyper parameters \({{\bf{\mu }}_{\bf{0}}}{\bf{,}}{{\bf{\lambda }}_{\bf{0}}}{\bf{,}}{{\bf{\alpha }}_{\bf{0}}}{\bf{,}}{{\bf{\beta }}_{\bf{0}}}\)
See all solutions

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