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Chapter 8: Q 2E (page 484)

Suppose that\({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\)form a random sample from the normal distribution with unknown mean 渭 and unknown standard deviation 蟽, and let\({\bf{\hat \mu }}\,\,{\bf{and}}\,\,{\bf{\hat \sigma }}\)denote the M.L.E.鈥檚 of 渭 and 蟽. For the sample size n = 17, find a value of k such that

\({\bf{Pr}}\left( {{\bf{\hat \mu > \mu + k\hat \sigma }}} \right){\bf{ = 0}}{\bf{.95}}\)

Short Answer

Expert verified

The value of k is -0.4365.

Step by step solution

01

Given information

Suppose that \({X_1},...,{X_n}\) from a random sample from the normal distribution with unknown mean is \(\mu \) and unknown standard deviation is \(\sigma \). And the sample size is\(n = 17\).

02

Finding the value of k

Since,

\(\hat \mu = {\bar X_n}\,\,and\,\,{\hat \sigma ^2} = \frac{{S_n^2}}{n}\)

It follows the t-distribution of the U has the t-distribution with n-1 degree of freedom.

It can be rewritten as,

\(U = \frac{{{n^{\frac{1}{2}}}\left( {{{\bar X}_n} - \mu } \right)}}{{{{\left( {\frac{{S_n^2}}{{n - 1}}} \right)}^{\frac{1}{2}}}}}\)

Then,

\(\begin{align}\Pr \left( {\hat \mu > \mu + k\hat \sigma } \right) &= \Pr \left( {\frac{{{{\bar X}_n} - \mu }}{{\hat \sigma }} > k} \right)\\ &= \Pr \left( {U > k{{\left( {n - 1} \right)}^{\frac{1}{2}}}} \right)\end{align}\)

Since U has the t-distribution with n-1 degrees of freedom and \(n = 17\)

Then,

\begin{align}\Pr \left[ {U > \,k{{\left( {n - 1} \right)}^{\frac{1}{2}}}} \right] = \,0.95\\\Pr \left[ {U\, > \,k{{\left( {17 - 1} \right)}^{\frac{1}{2}}}} \right] = \,0.95\\\Pr \left[ {U > k{{\left( {16} \right)}^{\frac{1}{2}}}} \right] = 0.95\\ \Pr \left[ {U > 4k} \right] = 0.95\end{align}

It is found from a table t-distribution with 16 degrees of freedom that,

\(\Pr \left( {U < 1.746} \right) = 0.95\)

Hence, by symmetry,

\(\Pr \left( {U > - 1.746} \right) = 0.95\)

So,

\(\begin{align}4k &= - 1.746\\k &= - \frac{{1.746}}{4}\\k &= - 0.4365\end{align}\)

Therefore, the value of k is -0.4365.

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Most popular questions from this chapter

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 \({{\bf{\mu }}_{\bf{0}}}{\bf{,}}{{\bf{\lambda }}_{\bf{0}}}{\bf{,}}{{\bf{\alpha }}_{\bf{0}}}{\bf{,}}{{\bf{\beta }}_{\bf{0}}}\)

Question:Suppose that a random variable X has the Poisson distribution with unknown mean 位 (位 > 0). Show that the only unbiased estimator of\({{\bf{e}}^{{\bf{ - 2\lambda }}}}\)is the estimator 未(X) such that 未(X) = 1 if X is an even integer and 未(X) = 鈭�1 if X is an odd integer.

We will draw a sample of size n = 11 from the normal distribution with the mean \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\). We will use a natural conjugate prior for the parameters \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\) from the normal-gamma family with hyperparameters \({{\bf{\alpha }}_{\bf{0}}}{\bf{ = 2,}}{{\bf{\beta }}_{\bf{0}}}{\bf{ = 1,}}{{\bf{\mu }}_{\bf{0}}}{\bf{ = 3}}{\bf{.5}}\,\,{\bf{and}}\,\,{{\bf{\lambda }}_{\bf{0}}}{\bf{ = 2}}\)

The sample yields an average of \(\overline {{{\bf{x}}_{\bf{n}}}} {\bf{ = 7}}{\bf{.2}}\,\,{\bf{and}}\,\,{{\bf{s}}_{\bf{n}}}^{\bf{2}}{\bf{ = 20}}{\bf{.3}}\)

a. Find the posterior hyperparameters.

b. Find an interval that contains 95% of the posterior distribution of \({\bf{\mu }}\).

Using the prior and data in the numerical example on nursing homes in New Mexico in this section, find (a) the shortest possible interval such that the posterior probability that \({\bf{\mu }}\) lies in the interval is 0.90, and (b) the shortest possible confidence interval for \({\bf{\mu }}\) which the confidence coefficient is 0.90.

Question: Consider the calorie count data described in Example7.3.10 on page 400. Now assume that each observation has the normal distribution with unknown mean \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\) given the parameter \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\). Use the normal-gamma conjugate prior distribution with prior hyper parameters

\({{\bf{\alpha }}_{\bf{0}}}{\bf{ = 1,}}{{\bf{\beta }}_{\bf{0}}}{\bf{ = 60,}}{{\bf{\mu }}_{\bf{0}}}{\bf{ = 0}}\,\,{\bf{and}}\,\,{{\bf{\lambda }}_{\bf{0}}}{\bf{ = 1}}\)The value of \({{\bf{s}}_{\bf{n}}}^{\bf{2}}\) is 2102.9.

a. Find the posterior distribution of \({\bf{\mu }}\,\,{\bf{and}}\,\,{\bf{\tau }}\)

b. Compute \({\bf{Pr(\mu > 1|x)}}{\bf{.}}\)
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