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?
The needed sample size is \(n \ge 16\)
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Suppose thatXhas thetdistribution withmdegrees of freedom(m >2). Show that Var(X)=m/(m−2).
Hint:To evaluate\({\bf{E}}\left( {{{\bf{X}}^{\bf{2}}}} \right)\), restrict the integral to the positive half of the real line and change the variable fromxto
\({\bf{y = }}\frac{{\frac{{{{\bf{x}}^{\bf{2}}}}}{{\bf{m}}}}}{{{\bf{1 + }}\frac{{{{\bf{x}}^{\bf{2}}}}}{{\bf{m}}}}}\)
Compare the integral with the p.d.f. of a beta distribution. Alternatively, use Exercise 21 in Sec. 5.7.
Suppose that\({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\)form a random sample from the normal distribution with unknown mean μ and unknown standard deviation σ, and let\({\bf{\hat \mu }}\,\,{\bf{and}}\,\,{\bf{\hat \sigma }}\)denote the M.L.E.’s of μ and σ. For the sample size n = 17, find a value of k such that
\({\bf{Pr}}\left( {{\bf{\hat \mu > \mu + k\hat \sigma }}} \right){\bf{ = 0}}{\bf{.95}}\)
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)\({\bf{Pr}}\left( {{\bf{0}}{\bf{.736 < \mu < 15}}{\bf{.680}}} \right)\).
For the conditions of Exercise 5, use the central limit theorem in Sec. 6.3 to find approximately the size of a random sample that must be taken so that \(P\left( {{\bf{|}}{{{\bf{\bar X}}}_{\bf{n}}} - {\bf{p|}}} \right) \ge 0.95\) whenp=0.2.
Suppose that each of two statisticians, A and B, independently takes a random sample of 20 observations from the normal distribution with unknown mean μ and known variance 4. Suppose also that statistician A finds the sample variance in his random sample to be 3.8, and statistician B finds the sample variance in her random sample to be 9.4. For which random sample is the sample mean likely to be closer to the unknown value of μ?
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