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Chapter 8: Q 8E (page 494)
At the end of Example 8.5.11, compute the probability that \(\left| {{{\bar X}_2} - \theta } \right| < 0.3\) given Z = 0.9. Why is it so large?
\(P\left( {\left| {{{\bar X}_2} - \theta } \right| < 0.3|Z = 0.9} \right) = 1\)
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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?
Suppose that\({{\bf{X}}_{\bf{1}}},{\bf{ \ldots }},{{\bf{X}}_{\bf{n}}}\)form a random sample from the normal distribution with unknown mean渭and known variance\({{\bf{\sigma }}^{\bf{2}}}\). Let\({\bf{\Phi }}\)stand for the c.d.f. of the standard normal distribution, and let\({{\bf{\Phi }}^{{\bf{ - 1}}}}\)be its inverse. Show that
the following interval is a coefficient\(\gamma \)confidence interval for渭if\({{\bf{\bar X}}_{\bf{n}}}\)is the observed average of the data values:
\(\left( {{{{\bf{\bar X}}}_{\bf{n}}}{\bf{ - }}{{\bf{\Phi }}^{{\bf{ - 1}}}}\left( {\frac{{{\bf{1 + }}\gamma }}{{\bf{2}}}} \right)\frac{{\bf{\sigma }}}{{{{\bf{n}}^{\frac{{\bf{1}}}{{\bf{2}}}}}}}{\bf{,}}{{{\bf{\bar X}}}_{\bf{n}}}{\bf{ + }}{{\bf{\Phi }}^{{\bf{ - 1}}}}\left( {\frac{{{\bf{1 + }}\gamma }}{{\bf{2}}}} \right)\frac{{\bf{\sigma }}}{{{{\bf{n}}^{\frac{{\bf{1}}}{{\bf{2}}}}}}}} \right)\)
Question:Suppose that a random variable X has the geometric distribution with an unknown parameter p (0<p<1). Show that the only unbiased estimator of p is the estimator \({\bf{\delta }}\left( {\bf{X}} \right)\) such that \({\bf{\delta }}\left( {\bf{0}} \right){\bf{ = 1}}\) and \({\bf{\delta }}\left( {\bf{X}} \right){\bf{ = 0}}\) forX>0.
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.
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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