Understand · explore · practise
Plan and document a binary-data investigation, audit a clearly synthetic example and evaluate sampling, seasonality and dependence before drawing a conclusion.
Before you startComplete binomial tests and model limitations.
01 / Make the investigation reproducible
Record population, event, baseline, alternative, level and sample design.
Then record every eligible observation using the same event definition. Keep real measurements distinguishable from constructed teaching examples.
These 20 binary observations were invented for this exercise. They are not measurements from a real device.
1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1
02 / Specify what one trial means
For a physical selector, one trial might be one activation under fixed settings; success might be output A. Define p as the long-run probability of A under those conditions. A baseline p₀=0.5 is a claim to test, not a probability estimated from the same sample. Decide in advance whether the question concerns an increase, decrease or either change.
Why is “test whether the device is good” too vague?
What binary outcome would count as success?
Define the event, conditions and reference probability. “Good” does not specify a parameter or a testable statistical claim.
Can you choose p₀ to equal your final sample proportion and call that an independent benchmark?
The baseline must have a defensible origin.
No. State where the baseline comes from, such as a design claim or an appropriate external benchmark. Choosing it from the same sample changes the question.
03 / Keep a complete observation log
For real data, record an identifier, date or trial number, measured outcome and its binary coding. Note exclusions with reasons fixed independently of the desired result. A missing observation is not automatically a failure. Do not stop early simply because the result has become significant.
A record is unreadable. Should it be coded zero?
Unknown is not the same as observed failure.
No. Record it as missing, explain the handling rule and use a justified eligible sample size.
Why not keep collecting until the first significant result?
Does that preserve the planned fixed-n test?
No. It introduces repeated opportunities to reject and changes the sampling rule. Choose and follow the fixed sample plan for this binomial investigation.
04 / Work through a transparent teaching dataset
Pause, replay or seek freely. The notes explain the same idea and stay in view.
The log has n=20 observations and x=16 successes
The observed proportion is 16/20=0.8.
Under H₀, X~B(20,0.5)
The upper inclusive tail is P(X≥16)=6196/1048576≈0.00590897.
0.00590897≤0.025
Reject H₀ in this constructed exercise: sufficient evidence of a change under the assumed model. This is not evidence about an actual device.
How many zeros are in the supplied sample?
Subtract successes from the sample size.
Four. There are 20 entries and 16 ones.
Why include count 16 in the tail?
It was the observed count.
The upper evidence event is the observed count or more extreme: 16,17,18,19,20.
05 / Use real records for a real-world conclusion
For an optional real investigation, save the original dataset and its source, date accessed, variables and units. State your sampling method and binary threshold before examining the test result. For a weather question, identify the station, dates and population period. Use records you can actually inspect; do not replace unavailable data with invented observations.
Give a reproducible definition of a weather success.
Specify threshold, variable and units.
For example, a valid daily rainfall record strictly above 2 mm at a named station during a defined period. State how traces and missing values are treated.
Why may a winter rainfall benchmark be unsuitable for a summer sample?
Consider a constant population success probability.
Seasonality can change the event probability. The baseline and sampled population may not be comparable.
06 / Test the model before trusting the result
Consecutive weather days may be dependent; pooling locations or seasons may mix different probabilities. A sample of volunteers may not represent the intended population. Explain these issues and limit the conclusion. A small numerical tail does not justify assumptions after the fact.
Why can pooling two locations with different climates undermine B(n,p)?
One p must apply to each trial.
The success chances may differ systematically between locations, so a common-p binomial model may not fit.
Does spacing observations apart automatically prove independence?
A design choice can reduce dependence without proving its absence.
No. Explain why independence is plausible and acknowledge remaining dependence or shared influences.
If real data do not reject, may you conclude that the baseline is correct?
Use the established evidence wording.
No. State insufficient evidence for the specified alternative at the chosen level, subject to the design and model limitations.
07 / Write a short auditable report
What should accompany the final probability?
Give the path from source to decision.
Source and sample plan, event definition, raw or accessible records, n and x, parameter and hypotheses, model assumptions, test convention and level, inclusive-tail calculation, contextual conclusion and limitations.
Can the supplied synthetic observations be described as measurements taken from a school experiment?
Were they measured?
No. Label them as constructed practice data. A real investigation requires actual records.
A real result contradicts your expectation. Should observations be removed to restore the expected conclusion?
Use the predefined eligibility rules.
No. Check recording and calculation errors, but retain eligible observations regardless of which conclusion they support. Document any justified correction.
08 / A credible investigation is more than a calculation
Define the binary event and population, justify the baseline, plan the sample, preserve the records and check assumptions. Use an appropriate inclusive tail and a preselected decision rule. Label synthetic practice honestly, and only make real-world claims from actual data.
Section 1 of 8 · Make the investigation reproducible