Selecting Inference Procedures
Interval or test?
- First decide the goal of the analysis.
- Estimate a value ("how big is it?") → a confidence interval.
- Judge a claim ("is it different from X?") → a significance test.
- The wording of the question tells you which.
Mean or proportion? How many?
- Proportion (a count of successes, a percentage) → a $z$-procedure.
- Mean (a measured quantity, an average) → a $t$-procedure.
- One group or two? And if two, independent or paired?
- Those three choices pin down exactly one procedure.
Implement it correctly
- State hypotheses (for a test) or the parameter (for an interval), in context.
- Check the conditions (random, 10%, and normal/large-counts).
- Compute the interval or the test statistic and p-value.
- Conclude — an interval's meaning, or reject / fail to reject.
Communicate clearly
- Show the reasoning: name the procedure, the conditions, the numbers.
- Tie the conclusion to the context — real groups, real units, real question.
- For a test: decision and a plain-language sentence about the alternative.
- For an interval: the range and what it says about the parameter.
Most mistakes are procedure-selection mistakes, not arithmetic. Ask in order: (1) interval or test? (goal), (2) proportion or mean? (data type → $z$ or $t$), (3) one sample, two independent, or paired? Get those three right and the formula follows; get one wrong and a perfectly-computed answer is still wrong.
"Is the mean recovery time different for two independent treatment groups?"
- Goal: judge a claim of difference → a test.
- Data: a measured time = a mean; two independent groups.
- → a two-sample $t$-test for a difference of means. (Had it asked "how much faster?", an interval.)
Choosing a procedure is three questions: interval or test (goal), proportion or mean ($z$ vs $t$), and one / two-independent / paired. Then implement it fully — hypotheses or parameter, conditions, computation, conclusion — and communicate the reasoning and result clearly, in context.
The z and t procedure family
Proportions use z; means use t — one or two samples each.
The question asks 'how much larger is the mean?' You should build a...
Estimating a value → interval; judging a claim → test.
Data are a MEASURED quantity (an average time). The procedure family is...
Measured average → mean → t-procedure.
Two independent treatment groups, judging a difference in means. The procedure is...
Independent groups + means + a claim → two-sample t-test.
Order the three questions that pick an inference procedure.
Goal, then data type, then sample structure.
Most inference mistakes are procedure-selection errors, not arithmetic errors.
Pick the right procedure and the formula follows.