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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 thatσ2=0.09. What numbercsatisfies Pr(T≤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

Prove that the distribution of\({\hat \sigma _0}^2\)in Examples 8.2.1and 8.2.2 is the gamma distribution with parameters\(\frac{n}{2}\)and\(\frac{n}{{2{\sigma ^2}}}\).

Suppose that\({{\bf{X}}_{\bf{1}}}{\bf{,}}...{\bf{,}}{{\bf{X}}_{\bf{n}}}\)form a random sample from the Poisson distribution with unknown mean θ, and let

\({\bf{Y = }}\sum\nolimits_{{\bf{i = 1}}}^{\bf{n}} {{{\bf{X}}_{\bf{i}}}} \).

a. Determine the value of a constant c such that the estimator\({{\bf{e}}^{{\bf{ - cY}}}}\)is an unbiased estimator of\({{\bf{e}}^{{\bf{ - \theta }}}}\).

b. Use the information inequality to obtain a lower bound for the variance of the unbiased estimator found in part (a).

Suppose that \({X_1},...,{X_n}\) form a random sample from the normal distribution with mean μ and variance \({\sigma ^2}\) . Assuming that the sample size n is 16, determine the values of the following probabilities:

\(\begin{align}a.\,\,\,\,P\left( {\frac{1}{2}{\sigma ^2} \le \frac{1}{n}\sum\limits_{i = 1}^n {{{\left( {{X_i} - \mu } \right)}^2} \le 2{\sigma ^2}} } \right)\\b.\,\,\,\,P\left( {\frac{1}{2}{\sigma ^2} \le \frac{1}{n}\sum\limits_{i = 1}^n {{{\left( {{X_i} - {{\bar X}_n}} \right)}^2} \le 2{\sigma ^2}} } \right)\end{align}\)

Suppose that X1,…….,Xn form a random sample from the Bernoulli distribution with unknown parameter p. Show that the variance of every unbiased estimator of (1-p)2 must be at least 4p(1-p)3/n.

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.
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