Chapter 9: Q2E (page 604)
2. Suppose that a random variable X has the F distribution with three and eight degrees of freedom. Determine the value of c such that Pr(X>c) = 0.975
The required value of c is 0069.
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Suppose that \({X_1},.....,{X_n}\)form a random sample from the \({\chi ^2}\) distribution with unknown degrees of freedom\(\theta \)\((\theta = 1,2,...)\), and it is desired to test the following hypotheses at a given level of significance\({\alpha _0}\left( {0 < {\alpha _0} < 1} \right)\):
\(\begin{array}{l}{H_0}:\;\;\;\theta \le 8,\\{H_1}:\;\;\;\theta \ge 9.\end{array}\)
Show that there exists a UMP test, and the test specifies rejecting \({H_0}\)if \(\sum\limits_{i = 1}^n {log} {X_i} \ge k\) for some appropriate constant\(k\).
1. Consider again the situation described in Exercise 11 of Sec. 9.6. Test the null hypothesis that the variance of the fusion time for subjects who saw a picture of the object is no smaller than the variance for subjects who did see a picture. The alternative hypothesis is that the variance for subjects who saw a picture is smaller than the variance for subjects who did not see a picture. Use a level of significance of 0.05.
An unethical experimenter desires to test the following hypotheses:
\(\begin{array}{*{20}{l}}{{{\bf{H}}_{\bf{0}}}{\bf{:}}}&{{\bf{\theta = }}{{\bf{\theta }}_{\bf{0}}}}\\{{{\bf{H}}_{\bf{1}}}{\bf{:}}}&{{\bf{\theta }} \ne {{\bf{\theta }}_{\bf{0}}}}\end{array}\)
She draws a random sample \({{\bf{X}}_{\bf{1}}}{\bf{, \ldots ,}}{{\bf{X}}_{\bf{n}}}\) from a distribution with the p.d.f. \(f(x\mid \theta )\), and carries out a test of size \(\alpha \). If this test does not reject \({{\bf{H}}_{\bf{0}}}\), she discards the sample, draws a new independent random sample of \(n\)observations, and repeats the test based on the new sample. She continues drawing new independent samples in this way until she obtains a sample for which \({{\bf{H}}_{\bf{0}}}\) is rejected.
a. What is the overall size of this testing procedure?
b. If \({{\bf{H}}_{\bf{0}}}\) is true, what is the expected number of samples that the experimenter will have to draw until she rejects \({{\bf{H}}_{\bf{0}}}\) ?
In Exercise 8, assume that Z=z is observed. Find a formula for thep-value.
Suppose that \({X_1},....,{X_{10}}\)form a random sample from a normal distribution for which both the mean and the variance are unknown. Construct a statistic that does not depend on any unknown parameters and has the \(F\)distribution with three and five degrees of freedom.
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