Question

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 \({{\bf{H}}_{\bf{0}}}\) ?

Short answer

(a)\(1\)

(b)\(\frac{1}{\alpha }\)

Step-by-step solution

To Find the overall size of this testing procedure

Consider the entire procedure that the experimenter undertakes, and consider each sample of \(n\) observations she takes as a Bernoulli trial, spewing out a success if \({H_0}\) is rejected, 0 otherwise. Then, if \({H_0}\) is true, and \(\alpha > 0\), her success probability if \(\alpha \), hence she obtains a success with probability 1 . Thus the overall size of the test is 1 (since the null will eventually be rejected almost surely).

The expected number of sample that the experimenter

The process downs to the number of trials required to obtain the first success in a sequence of Bernoulli trials. This thus follows a Geometric distribution with success probability \(\alpha \). Hence expected number of trials she needs to obtain to reject the null, is given by \(\frac{1}{\alpha }\).