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Chapter 8: Q1 E (page 472)

Suppose that we will sample 20 chunks of cheese in Example 8.2.3. Let\({\bf{T = }}\sum\limits_{{\bf{i = 1}}}^{{\bf{20}}} {{{\left( {{{\bf{X}}_{\bf{i}}}{\bf{ - \mu }}} \right)}^{\bf{2}}}{\bf{/20}}} \)wherexiis the concentration of lactic acid in theith chunk. Assume that蟽2=0.09. What numbercsatisfies Pr(T鈮�c)=0.9?

Short Answer

Expert verified

The value of c is 0.127854

Step by step solution

01

Given information

Xiis the concentration of lactic acid in theith chunk.

02

Calculating the value of c

The distribution of 20T/0.09 is \({\chi ^2}\) the distribution with 20 degrees of freedom

\(\begin{align}P\left( {T \le c} \right) &= 0.9\\P\left( {20T/0.09 \le 20c/0.09} \right) &= 0.9\\P\left( {{\chi ^2}\left( {20} \right) \le 20c/0.09} \right) &= 0.9\\20c/0.09 &= 28.41198\\c &= 0.127854\end{align}\)

So, the value of c is 0.127854

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Most popular questions from this chapter

Consider again the conditions of Exercise 19, and let\({{\bf{\hat \beta }}_{\bf{n}}}\)n denote the M.L.E. of 尾.

a. Use the delta method to determine the asymptotic distribution of\(\frac{{\bf{1}}}{{{{{\bf{\hat \beta }}}_{\bf{n}}}}}\).

b. Show that\(\frac{{\bf{1}}}{{{{{\bf{\hat \beta }}}_{\bf{n}}}}}{\bf{ = }}{{\bf{\bar X}}_{\bf{n}}}\), and use the central limit theorem to determine the asymptotic distribution of\(\frac{{\bf{1}}}{{{{{\bf{\hat \beta }}}_{\bf{n}}}}}\).

Suppose that two random variables\({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\)have the joint normal-gamma distribution with hyperparameters\({{\bf{\mu }}_{\bf{0}}}{\bf{ = 4,}}{{\bf{\lambda }}_{\bf{0}}}{\bf{ = 0}}{\bf{.5,}}{{\bf{\alpha }}_{\bf{0}}}{\bf{ = 1,}}{{\bf{\beta }}_{\bf{0}}}{\bf{ = 8}}\)Find the values of (a)\({\bf{Pr}}\left( {{\bf{\mu > 0}}} \right)\)and (b)\({\bf{Pr}}\left( {{\bf{0}}{\bf{.736 < \mu < 15}}{\bf{.680}}} \right)\).

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?

Show that two random variables \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\)cannot have the joint normal-gamma distribution such that

\({\bf{E}}\left( {\bf{\mu }} \right){\bf{ = 0}}\,\,{\bf{,Var}}\left( {\bf{\mu }} \right){\bf{ = 1,E}}\left( {\bf{\tau }} \right){\bf{ = }}\frac{{\bf{1}}}{{\bf{2}}}\,\,{\bf{and}}\,\,{\bf{Var}}\left( {\bf{\tau }} \right){\bf{ = }}\frac{{\bf{1}}}{{\bf{4}}}\)

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?
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