01 · Model assumptions
Name the binomial assumptions that matter in this sample.
Hint
Think about trials, outcomes and probabilities.
Worked solution
A fixed sample of 14, two categories per item, independent outcomes and a common mark probability.
Understand · explore · practise
Statistics review: tests and judgement
Cumulative original problems on complete binomial tests, discrete boundaries, actual levels, repeated testing and the limits of statistical conclusions.
Before you startHypothesis testing and the preceding statistics reviews.
01 / The planned rule determines the decision
Do not choose a method because it gives the result you want.
State the alternative and level before calculating.
These examples use at-most regions unless the question explicitly asks for closest tails. Numerical samples are constructed for practice.
Same observation, different planned testExplore
Under H₀, X~B(14,0.3). An observation of 8 has upper-tail probability 0.03146853. Two-sided choices use equal-tail at-most regions.
02 / Build a complete contextual test
Use a population parameter and the null probability.
A production process historically makes a particular mark with probability 0.3. An increase is suspected. A planned sample of 14 independent items contains eight marked items. Test at 5%.Worked example
Let p be the probability of a mark under current conditions
H₀:p=0.3; H₁:p>0.3.
Under H₀, X~B(14,0.3)
X counts marked items; the observed value is 8.
P(X≥8)≈0.03146853≤0.05
Reject H₀: sufficient evidence at 5%, under the model, of increased mark probability.
02 · Null probability
Why not use p=8/14 in the evidence calculation?
Hint
What hypothesis is the calculation testing?
Worked solution
The calculation assumes the claimed baseline p=0.3. The observed fraction is an estimate, not the null-model probability.
03 / Find the adjacent failing cutoff
A rare observation is not enough to identify the whole region.
Watch: changing the budget changes the decision
Pause, replay or seek freely. The notes explain the same idea and stay in view.
03 · Upper region at 5%
For B(14,0.3), P(X≥7)≈0.09328189 and P(X≥8)≈0.03146853. Find the upper critical region.
Hint
Use the smallest qualifying count.
Worked solution
X≥8 within 0,…,14. Eight meets the 0.05 limit, while seven fails.
04 · Actual level
State the actual significance of that upper test.
Hint
Sum the null mass over its rejection region.
Worked solution
P(X≥8)≈0.03146853, or 3.146853%. It is below the nominal 5%.
05 · Stricter level
Also P(X≥9)≈0.00828852. Find the 1% upper critical region and classify observation 8.
Hint
Compare both neighbouring candidates with 0.01.
Worked solution
The region is X≥9. Eight does not reject at 1%, since its tail exceeds 0.01.
04 / Allocate both tails before checking the sample
The upper observed count does not remove the lower tail.
For the same B(14,0.3) null model, F(0)≈0.00678223 and F(1)≈0.04747562.
06 · Two-sided 5% region
Construct the nominal 5% equal-tail at-most region using those lower tails and the upper tails above.
Hint
Each tail must be at most 0.025.
Worked solution
Lower region X=0, since F(1)>0.025. Upper region X≥9, since P(X≥8)>0.025. The full region is X=0 or X≥9.
07 · Achieved two-sided level
Find its actual significance.
Hint
Add the two disjoint tails.
Worked solution
Approximately 0.00678223 + 0.00828852 = 0.01507075, or 1.507075%.
08 · Compare decisions
Why does observation 8 reject the one-sided 5% test but not this two-sided 5% test?
Hint
The alternatives allocate the probability budget differently.
Worked solution
Its upper-tail probability 0.03146853 is below 0.05 but above 0.025. The test direction must have been chosen before observing eight.
05 / Read the instruction carefully
Closest and at-most solve different boundary problems.
09 · Explicit closest lower tail
For target 0.025, which is closer: F(0)=0.00678223 or F(1)=0.04747562?
Hint
Compare absolute differences.
Worked solution
F(0) is closer: about 0.01821777 away, compared with 0.02247562 for F(1).
10 · Explicit closest upper tail
Compare upper tails at 8 and 9 with target 0.025. Which is closer?
Hint
Compare distances 0.00646853 and 0.01671148.
Worked solution
The tail at 8 is closer, although it exceeds 0.025. Only use that choice when the instruction explicitly permits closest tails.
11 · Closest total
Find the actual total for the closest-tail region just obtained.
Hint
Combine X=0 and X≥8.
Worked solution
Approximately 0.00678223 + 0.03146853 = 0.03825076, or 3.825076%. Report this achieved value rather than assuming it equals 5%.
06 / Do not substitute nominal for actual in repeated tests
The false-rejection calculation assumes true nulls.
12 · Repeated independent tests
Two independent true-null tests each have actual level a=0.03146853243564. Write the probability at least one rejects.
Hint
Complement both nonrejections.
Worked solution
1−(1−a)²≈0.0619468. Use the actual level, not 0.05.
13 · Unsupported multiplication
What if both tests share most of their observations?
Hint
Check whether their decisions can be treated independently.
Worked solution
The multiplication is not justified without an independence argument or a suitable joint model. Shared data can induce dependence.
07 / A conclusion must respect the study design
Keep the parameter, data and causal claim separate.
14 · Sampling scope
If all 14 items come from one unusual shift, what limits a conclusion about all current production?
Hint
Consider representativeness and common conditions.
Worked solution
The sample may not represent other shifts, and conditions or mark probabilities may differ. State this scope limitation.
15 · Nonrejection
Write a sound conclusion for observation 8 under the two-sided 5% at-most test.
Hint
Use insufficient evidence, not proof of equality.
Worked solution
There is insufficient evidence at nominal 5%, under the stated rule and model, that the population mark probability differs from 0.3.
16 · Practical meaning
Does rejection alone establish a large or causal increase?
Hint
What else would be needed?
Worked solution
No. Consider effect size, uncertainty and the study design. A test supplies statistical evidence within a model, not causal proof or a measure of practical importance.
08 / A complete statistical answer includes judgement
Use the calculation to support a clearly scoped conclusion.
Define the parameter, justify the model, fix the alternative and level, use inclusive probabilities and check neighbouring boundaries. Distinguish at-most from closest instructions, report actual levels when needed and acknowledge material sampling or dependence limitations.
Section 1 of 8 · The planned rule determines the decision