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Chapter 9: Q5E (page 548)

Suppose that X1,鈥�.., Xn forms a random sample from the normal distribution with unknown mean 碌 and unknown variance 蟽2. Classify each of the following hypotheses as either simple or composite:

a) H0 : 碌 = 0 and 蟽=1

b) H0 : 碌 > 3 and 蟽<1

c) H0 : 碌 = -2 and 蟽2<5

d) H0 : 碌 = 0

Short Answer

Expert verified

a.Given hypothesis is a simple hypothesis.

b.Given hypothesis is a composite hypothesis.

c.Given hypothesis is a composite hypothesis.

d.Given hypothesis is a composite hypothesis.

Step by step solution

01

Given Information

The parameter space for the parameters of N( 碌, 蟽2 ) is given by R脳R+ (as both the mean and variance are unknown). A hypothesis is simple if it contains a single point, and composite otherwise.

02

Determine if the following assumptions are simple or composite

a. Given hypothesis is H0 : 碌 = 0 and 蟽=1

The space corresponding to the hypothesis is {(0,1)}. Thus this hypothesis is a simple hypothesis.

b. Given hypothesis isH0 : 碌 > 3 and 蟽<1

The space corresponding to the hypothesis is (3,鈭�) 脳 (0,1). Hence clearly this hypothesis is a composite hypothesis.

c. Given hypothesis isH0 : 碌 = -2 and 蟽2<5

The space corresponding to this hypothesis is {-2} 脳(0,鈭�5).Hence clearly this hypothesis is a composite hypothesis.

d. Given hypothesis isH0 : 碌 = 0. The space corresponding to this hypothesis is given by{0} 脳 R+. Hence clearly this hypothesis is a composite hypothesis.

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Most popular questions from this chapter

Consider the situation described immediately before Eq. (9.1.12). Prove that the expression (9.1.12) equals the smallest伪0such that we would reject H0 at level of significance伪0.

Suppose that the 12 observations \({X_1},....,{X_{12}}\)form a random sample from the normal distribution with unknown mean \(\mu \)and unknown variance\({\sigma ^2}\). Describe how to carry out a \(t\)test of the following hypotheses at the level of significance\({\alpha _0} = 0.005\):

\(\begin{array}{l}{H_0}:\;\;\;\mu \ge 3 \\{H_1}:\;\;\;\mu < 3\end{array}\)

Consider again the conditions of Exercise 4, and suppose that伪(未)is required to be a given value伪0 (0 < 伪0< 1). Determine the test procedure未for which尾 (未)will be a minimum, and calculate this minimum value.

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\(k\).

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