Understand · explore · practise
Independent mixed problems on hypotheses, exact binomial tails, critical regions, actual levels, repeated tests and justified contextual conclusions.
Before you startThe preceding hypothesis-testing lessons.
01 / Choose the method yourself
Define → model → choose tail → calculate → conclude.
Keep the chosen alternative and significance level fixed. These questions use at-most critical regions unless a closest-tail instruction is stated.
02 / Parameters, events and direction
A training programme is suspected to raise a pass probability from 0.6. Define p and state hypotheses.
The parameter is a population probability.
Let p be the probability a pupil from the defined target population passes under the programme. H₀:p=0.6; H₁:p>0.6.
The same baseline pass probability is 0.6. Define q as failure probability and translate the improvement claim.
q=1−p.
H₀:q=0.4; H₁:q<0.4. Improvement raises passing and lowers failing.
A selector is claimed to produce a red output with probability 0.25. A technician suspects a change. State the alternative.
Both increases and decreases count.
H₁:p≠0.25, where p is the red-output probability. H₀:p=0.25.
In 10 trials there are 8 red outputs. Distinguish p, X and x.
Separate unknown probability, random count and observation.
p is the population red probability. X is the random red count in 10 trials. The observed value is x=8; the sample proportion is 0.8.
03 / Select inclusive evidence events
For X~B(10,0.5), an upper-tail test observes 8. Write the evidence event and its cumulative complement.
The observed count belongs in the tail.
X≥8, with probability 1−P(X≤7). The probability is 0.0546875.
Under H₀, X~B(8,0.25). Against p>0.25, observe 5. The upper tail is 0.02729797. Conclude at 5%.
Compare with the full one-sided 0.05 level.
Reject H₀. There is sufficient evidence at 5% under the model that the red probability exceeds 0.25.
For the same test, observing 4 gives upper tail 0.11381531. Conclude at 5%.
Use the inclusive tail, not P(X=4).
Do not reject H₀. There is insufficient evidence at 5% of an increased red probability.
Under H₀, Y~B(12,0.75), against p<0.75, observe 5. Its lower tail is 0.01425278. Conclude at 5%.
Small counts support this alternative.
Reject H₀. There is sufficient evidence at 5% that the population success probability has decreased.
04 / Justify both sides of a critical value
Pause, replay or seek freely. The notes explain the same idea and stay in view.
For B(10,0.5), P(X≥8)=0.0546875 and P(X≥9)=0.01074219. Find the 5% upper critical region and actual level.
Choose the smallest qualifying upper cutoff.
X≥9, or X=9 or 10. Actual level is 0.0107421875, about 1.0742%. Eight fails the 0.05 requirement.
For B(12,0.75), P(Y≤5)=0.01425278 and P(Y≤6)=0.05440223. Find the 5% lower region.
Choose the largest qualifying lower cutoff.
Y≤5. Five meets 0.05, but six exceeds it. The actual level is about 1.4253%.
An observation Y=6 occurs in question 10. State the decision and explain.
Use the region just constructed.
Do not reject H₀: six is outside Y≤5. Its lower tail also exceeds 0.05, agreeing with the region method.
For B(4,0.1), P(X=0)=0.6561. Can a 5% lower-tail rule reject any possible count?
All nonempty lower tails contain zero.
No. Even the smallest nonempty lower tail exceeds 0.05. The lower rejection region is empty.
05 / Two-sided rules and achieved levels
What tail budget belongs to each side of a nominal 10% equal-tail test?
Halve the decimal level.
0.05, or 5%, per tail.
For B(10,0.5), lower probabilities at 1 and 2 are 0.01074219 and 0.0546875. Find the nominal 10% equal-tail at-most region.
Use symmetry and reject the candidate exceeding 0.05.
X≤1 or X≥9. Actual level is 22/1024=0.021484375, about 2.1484%.
Now explicitly choose each tail closest to 0.05. What changes?
Compare distances on both sides of the target.
Choose X≤2 or X≥8 because 0.0546875 is closer to 0.05. Actual level is 112/1024=0.109375, above 10%. This is not an at-most 10% rule.
For an observed count 8, compare the decisions in questions 14 and 15.
Check each region.
Eight does not reject under the at-most rule but does under the explicitly closest-tail rule. State which construction was requested.
06 / Actual level and repeated opportunities
A fixed test has actual level 0.03. What does that probability describe?
State the condition.
It is the probability the test rejects when H₀ is true, under its model. It is not the probability H₀ is true after rejection.
Two independent tests both have true nulls and actual level 0.03. Find the probability at least one rejects.
Complement no rejection.
1−0.97²=0.0591.
For those two tests, find the probability both reject.
Multiply the independent rejection probabilities.
0.03²=0.0009. This differs from at least one rejection.
Why cannot those products be assumed for two tests using the same observations?
The decisions can be dependent.
Independence is not guaranteed. A suitable joint model or additional justification is needed.
07 / Conclude without overclaiming
Repair “the result is not significant, so the original probability is definitely correct”.
Use insufficient evidence.
There is insufficient evidence at the stated level for the chosen alternative, under the model. The null probability has not been proved.
A researcher chose a two-sided claim, then switches to an upper-tail test after seeing a high count. What is wrong?
The decision rule must precede the observation.
Selecting the direction from the result changes the planned test and can exaggerate evidence. Keep the prespecified alternative or acknowledge a different exploratory analysis.
Twenty adjacent seeds share one contaminated tray. What binomial assumption may fail?
A common cause can link outcomes.
Independence. Also consider whether conditions justify the same germination probability for each seed.
A worked example uses constructed observations. What may its conclusion establish about a real population?
No field measurements were supplied.
Nothing empirically about a real population. It demonstrates the method conditionally on the constructed data and assumed model.
08 / Use errors to choose the next lesson
If hypotheses or direction were unclear, revisit test tails. If a boundary was wrong, revisit inclusive binomial tails and critical regions. If the arithmetic was correct but the conclusion overclaimed, revisit conclusions and model limitations. Keep practising with the solutions closed first.
Section 1 of 8 · Choose the method yourself