Decision rule:

• If the p-value is less than the level of significance (α), reject the null hypothesis.
• Otherwise, fail to reject the null hypothesis.

In this context, the p-value of 0.036 is less than the level of significance of 0.05. Hence, the null hypothesis is rejected at α=0.05. Therefore, the given statement is true.

Level of significanceα:

Type I error is defined as the probability of rejecting the null hypothesis, when it is actually true.

In a test of hypothesis, α is the probability that the null hypothesis is true. Therefore, the given statement is false.

In a test of hypothesis, in order to compare the two populations, the pooled variance can also be small that leads to the greater value of the test statistic as pooled variance. Therefore, the given statement is false.

In a test of hypothesis, power is the probability that the null hypothesis is rejected when θ=θ*. Therefore, the given statement is true.

Normally, the power of test is computed by assuming that the null hypothesis is false and computed for the specific values of Ha.

Based on the decision rule provided in Part (a), when 0.01<p-value<0.025 can be either greater or smaller than the level of significance of 0.02. The null hypothesis is not rejected for the given scenario. Therefore, the given statement is false.

In this context, the given statement is false. This is because the uniformly most powerful test may have the highest power against all other α-level tests, and for some values of θa in Ha. In such case, the power of test is equal to 1−β(θa).

Likelihood-ratio test:

A likelihood-ratio test of H0:Θ∈Ω0 versus Ha:Θ∈Ω^a denote λ as the test statistic and the rejection region as λ≤k.

The test statistic λ is denoted as follows:

λ=L(Ω^0)L(Ω^)=maxΘ∈Ω^0L(Ω^0)maxΘ∈Ω^L(Ω^)

In this scenario, the provided statement is false, it is because L(Ω^)=max{L(Ω^0),L(Ω^1)}.

Based on Theorem 10.2, let Y1,Y2,...Yn have a joint likelihood function L(Θ). Denote r0 as the number of free parameter that specifies H0:Θ∈Ω0 and denote r as the number of free parameter that specifies Θ∈Ω. Then, for the large sample of size n, −2ln(λ) (always positive) has approximately a χ2 distribution with r0−r df.

Based on the theorem, the provided statement is True.