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An unethical experimenter desires to test the following
Chapter 9, Problem 17(choose chapter or problem)
An unethical experimenter desires to test the following hypotheses:
\(\begin{array}{l} H_{0}: \quad \theta=\theta_{0}, \\ H_{1}: \quad \theta \neq \theta_{0} . \end{array}\)
She draws a random sample \(X_{1}, \ldots, X_{n}\) 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 \(H_{0}\), 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 \(H_{0}\) is rejected.
a. What is the overall size of this testing procedure?
b. If \(H_{0}\) is true, what is the expected number of samples that the experimenter will have to draw until she rejects \(H_{0}\)?
Questions & Answers
QUESTION:
An unethical experimenter desires to test the following hypotheses:
\(\begin{array}{l} H_{0}: \quad \theta=\theta_{0}, \\ H_{1}: \quad \theta \neq \theta_{0} . \end{array}\)
She draws a random sample \(X_{1}, \ldots, X_{n}\) 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 \(H_{0}\), 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 \(H_{0}\) is rejected.
a. What is the overall size of this testing procedure?
b. If \(H_{0}\) is true, what is the expected number of samples that the experimenter will have to draw until she rejects \(H_{0}\)?
Step 1 of 3
Given data:
The random variables \(x_{1} \ldots \ldots \ldots \ldots x_{n}\) are given to be normally distributed with mean \(\mu\) and variance \(\sigma^{2}\) , where these two are unknown, \(\bar{X}\) is the mean, and \(\alpha\) is the sample size.
The null hypothesis to be tested can be written as follows:
\(H_{0}: \theta=\theta_{0}\)
The alternative hypothesis is given as follows:
\(H_{1}: \theta \neq \theta_{0}\)
The level of significance is \(\alpha\). If the test does not reject \(H_{0}\), discard the sample and draw a new random sample of \(n\) observations.