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Distinguish a population probability from a sample count, write hypotheses in context and state a binomial model under the null hypothesis.
Before you startBinomial model assumptions and basic probability notation.
01 / What are we testing?
Define the parameter before writing a hypothesis.
Let p be the probability of a clearly specified success in the population or process being studied.
A hypothesis is a statement about that underlying parameter. The observed proportion is evidence used to investigate it; it is not automatically the true value of p. Different random samples can give different proportions even when p stays fixed.
Choose a case and identify the population probability, trial total and observed count before revealing the answer.
02 / Keep p, n, X and x separate
Pause, replay or seek freely. The notes explain the same idea and stay in view.
p = probability a parcel from the relevant service arrives on time
Define the population and what counts as success.
n=25; X = number on time in the sample of 25
Before collecting the sample, X is random.
The observed value is x=18; the sample proportion is 18/25=0.72
This observation does not itself establish that the population probability is 0.72.
A seed supplier claims a germination probability of 0.9. In 40 trials, 33 seeds germinate. Define p and X.
p is a probability; X is a count with a stated sample size.
p is the probability that a seed from the specified batch germinates under the stated conditions. X is the number that germinate among the 40 tested seeds.
In the seed example, give n, x and the observed proportion.
Divide the count by the number tested.
n=40, x=33 and the observed proportion is 33/40=0.825.
03 / State the null hypothesis
For this course, write H₀: p=p₀.
p₀ is the specified baseline probability. We calculate what samples would look like if that baseline were true.
For the delivery example, H₀: p=0.8. This is a working assumption for the test, not a fact established by the sample. Do not replace 0.8 with 18/25 when calculating probabilities under H₀.
A student writes H₀: X=0.8 for the delivery example. Correct it.
X counts parcels; p is a probability.
H₀: p=0.8. The hypothesis concerns the population probability, not the random count.
Why is H₀: p=0.72 not the stated null in that example?
Where did 0.72 come from?
0.72 is the observed proportion. The stated claim being tested supplies the baseline 0.8. Replacing it with the sample estimate changes the question.
04 / State the alternative claim
If the service is suspected of becoming less reliable, use H₁: p<0.8. If improvement is suspected, use H₁: p>0.8. If any change is being investigated, use H₁: p≠0.8. Choose this claim before using the observation to decide which direction looks convenient.
A machine historically produces faulty components with probability 0.06. A modification is intended to reduce faults. Define p and write H₀ and H₁.
Success can mean a fault; it need not be desirable.
Let p be the new probability a component is faulty. H₀: p=0.06; H₁: p<0.06.
The same machine is being checked for any change in its fault probability. Write H₁.
Both increases and decreases matter.
H₁: p≠0.06. The null remains p=0.06.
05 / State the distribution under H₀
Check a fixed sample of 25 and an on-time/late classification
The success event must be unambiguous.
Assume independent outcomes and the same on-time probability
Shared delays or different routes may make these assumptions doubtful.
Under H₀, X~B(25,0.8)
The observed count remains 18. We do not write X~B(18,0.72).
For the germination example, state the null distribution, assuming independent seeds with a common germination probability.
Use the trial total and claimed baseline.
Under H₀: p=0.9, X~B(40,0.9).
Why could testing 25 parcels carried on one delayed vehicle undermine a binomial model?
Think about shared causes.
One delay can affect many parcels together, so their on-time outcomes may not be independent. A fixed sample size alone does not justify binomial probabilities.
06 / The test statistic summarises the sample
Before sampling, X is the random test statistic; after sampling, x is its observed value. With fixed n, the sample proportion X/n contains the same count information, but the binomial distribution applies to X, not directly to X/n. State the chosen count clearly.
Ten independent spins produce four blue results. Let p be the blue probability. State X and x.
Include the number of spins in the definition.
X is the number of blue results in ten spins; x=4. The observed proportion is 0.4.
A sample of 60 has an observed success proportion of 0.35. What count should be used in a binomial calculation?
Multiply by n.
x=60×0.35=21. Under a null p=p₀, calculate using X~B(60,p₀).
07 / A difference is not yet a conclusion
The delivery sample proportion 0.72 is below 0.8, but a test must still assess how unusual the relevant sample results are under H₀. Later lessons calculate that tail probability and compare it with a stated significance level. A nonrejection will mean insufficient evidence against H₀, not proof that the claim is true.
A learner says “18/25 is less than 0.8, so the claim is false.” What is missing?
Random samples need not match the baseline exactly.
They have not assessed sampling variation under the null model or used a significance threshold. A difference alone is not enough to establish statistical evidence.
A trial continues until the first successful attempt. Can the number of attempts be modelled as B(n,p) with a fixed n?
Which binomial requirement fails?
No. The number of attempts is random and measures a waiting count, not successes in a fixed trial total.
Can a spinner with three possible colours support a binomial test about blue?
How many categories does the chosen success event create?
Yes, if the number of spins is fixed and spins are independent with a common blue probability. Classify blue as success and all other colours as failure.
Why is “p is the probability of success” often too vague in a contextual answer?
State whose outcomes and what event are being counted.
It does not identify the process, population or success event. For example, p is the probability that a parcel sent by the specified service arrives by the promised time.
08 / Write the setup before calculating
Define p in context. State H₀ and H₁ using p. Define the success count X and record n and x separately. State the binomial distribution under H₀ only when its assumptions are justified. The next lessons turn this setup into a test.
Section 1 of 8 · What are we testing?