Concept explainers
True or False.
a If the p-value for a test is .036, the null hypothesis can be rejected at the α = .05 level of significance.
b In a formal test of hypothesis, α is the probability that the null hypothesis is incorrect.
c If the p-value is very small for a test to compare two population means, the difference between the means must be large.
d Power (θ*) is the probability that the null hypothesis is rejected when θ = θ*.
e Power (θ) is always computed by assuming that the null hypothesis is true.
f If .01 < p-value < .025, the null hypothesis can always be rejected at the α = .02 level of significance.
g Suppose that a test is a uniformly most powerful α-level test regarding the value of a parameter θ. If θa is a value in the alternative hypothesis, β(θa) might be smaller for some other α-level test.
h When developing a likelihood ratio test, it is possible that
i −2 ln(λ) is always positive.
a.
Check whether the given statement as true or false.
Answer to Problem 115SE
True.
Explanation of Solution
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
b.
Pinpoint the given statement as true or false.
Answer to Problem 115SE
False.
Explanation of Solution
Level of significance
Type I error is defined as the probability of rejecting the null hypothesis, when it is actually true.
In a test of hypothesis,
c.
State whether the given statement as true or false.
Answer to Problem 115SE
False.
Explanation of Solution
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.
d.
Define whether the given statement as true or false.
Answer to Problem 115SE
True.
Explanation of Solution
In a test of hypothesis, power is the probability that the null hypothesis is rejected when
e.
Delineate whether the given statement as true or false.
Answer to Problem 115SE
False.
Explanation of Solution
Normally, the power of test is computed by assuming that the null hypothesis is false and computed for the specific values of
f.
Find whether the given statement as true or false.
Answer to Problem 115SE
False.
Explanation of Solution
Based on the decision rule provided in Part (a), when
g.
Discover whether the given statement as true or false.
Answer to Problem 115SE
False.
Explanation of Solution
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
h.
Determine whether the given statement as true or false.
Answer to Problem 115SE
False.
Explanation of Solution
Likelihood-ratio test:
A likelihood-ratio test of
The test statistic
In this scenario, the provided statement is false, it is because
i.
Determine the given statement as true or false.
Answer to Problem 115SE
True.
Explanation of Solution
Based on Theorem 10.2, let
Based on the theorem, the provided statement is True.
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Chapter 10 Solutions
Mathematical Statistics with Applications
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