Two analysts both test H₀: μ = 0 vs H₁: μ = 5 at α = 0.05. Analyst A uses the likelihood ratio test. Analyst B invents a different test that also maintains exactly 5% false positives. Which test has higher power?
AAnalyst B's test — novel approaches can outperform classical methods
BAnalyst A's test — the Neyman-Pearson lemma guarantees no test of size α can have higher power
CThey are equally powerful, since both maintain α = 0.05
DIt depends on the sample size — the lemma only applies asymptotically
The Neyman-Pearson lemma is not a rule of thumb — it is a proof. Among all tests that keep Type I error at exactly α, the likelihood ratio test is most powerful: it minimizes Type II error (maximizes the probability of correctly rejecting H₀ when H₁ is true). No other test of the same size can do better. Analyst B's test is either equivalent to the LR test in its rejection region or strictly less powerful.
Question 2 Multiple Choice
When testing H₀: p = 0.5 vs H₁: p = 0.7 in n = 10 coin flips, the Neyman-Pearson optimal rejection region is 'reject when the number of heads k ≥ c.' Why does this follow from the likelihood ratio?
AThe sample mean is always the optimal test statistic for binomial hypotheses
BThe likelihood ratio L(0.7|k)/L(0.5|k) is a monotonically increasing function of k, so large k is the strongest evidence for H₁
CHead counts are sufficient statistics, and sufficient statistics always define optimal rejection regions
DThe p-value formula for binomial tests is defined in terms of head counts by convention
The likelihood ratio L(0.7|k)/L(0.5|k) = (0.7/0.5)^k · (0.3/0.5)^(10−k) increases as k increases — more heads make H₁ more plausible relative to H₀. The NP lemma says to reject when this ratio exceeds threshold k, which here reduces to rejecting when the head count exceeds some critical value c. The optimal test statistic and rejection region emerge naturally from maximizing the likelihood ratio, not from convention.
Question 3 True / False
The Neyman-Pearson lemma applies only to simple vs. simple hypothesis tests where both H₀ and H₁ specify a single parameter value.
TTrue
FFalse
Answer: True
The basic NP lemma is stated for simple hypotheses: H₀: θ = θ₀ vs H₁: θ = θ₁. The extension to composite alternatives — where H₁ covers a range of values — leads to the concept of Uniformly Most Powerful (UMP) tests, which are most powerful against every value in the alternative simultaneously. Not all testing problems admit a UMP test. Understanding the NP lemma as the simple-case foundation is essential before extending it.
Question 4 True / False
Reducing the significance level α (from 0.05 to 0.01, say) while keeping everything else constant will increase the power of a Neyman-Pearson test.
TTrue
FFalse
Answer: False
Power and significance level are in fundamental tension. Reducing α tightens the rejection region — you require stronger evidence to reject H₀. This means you will correctly detect H₁ less often: Type II error increases and power decreases. The Neyman-Pearson framework makes this tradeoff explicit: fixing α is the constraint, and the lemma maximizes power *given* that constraint. You cannot decrease α and increase power simultaneously without additional information (such as a larger sample).
Question 5 Short Answer
Why is the likelihood ratio — rather than some other function of the data — the key quantity in the Neyman-Pearson lemma?
Think about your answer, then reveal below.
Model answer: The likelihood ratio L(θ₁|X)/L(θ₀|X) directly measures how much better the data supports H₁ compared to H₀. Rejecting H₀ when this ratio is large concentrates the rejection region on exactly those outcomes most consistent with H₁, maximizing the probability of correct rejection while using the full α budget of Type I error. Any other rejection region would either miss some of the most H₁-consistent outcomes or include some H₀-consistent ones.
The likelihood ratio is the sufficient statistic for the binary comparison between two simple hypotheses — it captures everything in the data relevant to distinguishing θ₀ from θ₁. Ranking outcomes by their likelihood ratio gives the ordering that maximizes power: the outcomes most likely under H₁ relative to H₀ should be rejected first. The NP lemma formalizes this intuition into a proof of optimality.