- Distinguish sample spread from uncertainty in a sample mean.
- Estimate a proportion and its standard error.
- State the assumptions and interpretation of a confidence interval.
Concepts
A sample mean is also random
For n independent identically distributed observations with mean μ and finite variance σ², the sample mean has expectation μ and variance σ²/n. Its standard deviation, called its standard error, is σ/√n. Four times as much data halves this standard error; it does not quarter it. Dependence can invalidate the independent-sample calculation.
Estimate a chance with a proportion
For n independent Bernoulli observations, the fraction of successes estimates p. Its true standard error is √[p(1−p)/n]. Replacing p by the observed proportion gives a plug-in estimate. A normal approximation is useful with enough expected successes and failures, but becomes unreliable near 0 or 1 or with very small samples.
Separate uncertainty from bias
Under suitable normal or large-sample assumptions, an approximate 95% confidence interval is estimate ±1.96 standard errors. The 95% refers to long-run coverage of the procedure over repeated samples; it is not a frequentist probability assigned to a fixed parameter after seeing one interval. More observations can reduce sampling noise while leaving selection bias untouched.
Examples
Worked example 1
Precision of a sample mean
Sixty-four independent observations from a normal population have sample mean 12. The population standard deviation is known to be 4. Find the standard error and a 95% confidence interval for the population mean.
Scale the population spread
The standard error is 4/√64=4/8=0.5. This describes uncertainty in the sample mean, not spread of individual observations.Find the margin
Using the standard normal 97.5th percentile, approximately 1.96, the margin is 1.96×0.5=0.98.Center the interval
The interval is [12−0.98,12+0.98]=[11.02,12.98]. Normality and known population spread justify the normal-based procedure here.
Worked example 2
Estimating a success probability
A fixed process produces 96 successes in 400 independent trials. Estimate p and its standard error, and form an approximate 95% normal interval.
Compute the observed proportion
The estimate is 96/400=0.24. Both observed success and failure counts are comfortably away from zero.Plug into the standard-error formula
The estimated standard error is √[0.24×0.76/400]≈0.02135.State the approximation
The margin is about 1.96×0.02135≈0.04185, giving approximately [0.198,0.282]. This is a large-sample approximation under the independent, fixed-probability model.
Explore the idea · unscored
Watch an estimate settle with repeated trials
Each trial is an illustrative success or failure with the same fixed probability. The running proportion estimates that probability. Each numbered sequence is repeatable; increasing its length preserves the earlier trials.
Repeatable sequence 1
Solid line: observed success proportion. Dashed line: true probability. This is one repeatable simulation, not a scored run or real learner performance.
Observed proportion after 200 trials36.0%
72 successes ÷ 200 trials. Model standard error √[p(1−p)/n] = 3.24 percentage points. More data reduces typical sampling variation; this particular path need not get closer at every step.
Read the trial checkpoints
| Trials | Successes | Proportion |
|---|---|---|
| 1 | 0 | 0.0000 |
| 10 | 2 | 0.2000 |
| 20 | 6 | 0.3000 |
| 50 | 19 | 0.3800 |
| 100 | 35 | 0.3500 |
| 200 | 72 | 0.3600 |
Concept checks
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Key results
- Standard error describes an estimator's sampling variation.
- More data helps at a square-root rate under independence.
- Label approximations and check their assumptions.
- Precision does not guarantee lack of bias.
Practice problems
Solve related problems or try a timed probability drill.