Estimate a mean when population spread is unknown

Try

A sample of 36 independent process durations has mean 50 minutes and sample standard deviation 12 minutes. The population standard deviation is unknown. Predict the uncertainty of the sample mean, then explain why the uncertainty in estimating spread also matters when choosing a critical reference distribution.

Predict

Would dividing 12 by 36 give the standard error? Averaging reduces variance by n under the independent model, so standard deviation reduces by √n. Also predict whether an interval for the mean is wide enough to describe where an individual future duration will fall.

Experiment

Estimated SE is s/√n=12/6=2 minutes. For a 95% interval, use the supplied t critical value 2.030 with df=35. The margin is 2.030·2=4.06, giving [45.94,54.06] minutes.

Draw the interval around 50 and label its margin separately from the sample SD of 12. These two lengths describe different variability. State the sampling and distributional assumptions before interpreting the endpoints; no numerical formula can establish that the observations were independently or representatively collected.

Understand

For a sample of size n with mean x̄ and sample SD s, the usual one-mean t interval is x̄±t* s/√n, with df=n−1. The t critical value depends on confidence and degrees of freedom. At small df, its heavier tails account for additional uncertainty from estimating population spread. As df grows, the t reference approaches the standard normal reference.

Under independent observations from a normal population with common mean and positive variance, the classical standardized mean has an exact t distribution. For nonnormal populations, the method may be a useful approximation when the mean’s sampling distribution is adequately normal. Small samples with severe skew or outliers deserve particular caution; there is no universal sample-size cutoff that fixes every shape.

Inspect a plot and the collection process. The mean and SD are sensitive to unusual observations. Investigate a suspicious value’s provenance, but do not delete a valid extreme simply to improve a p-value or narrow an interval. A large sample does not cure dependence, convenience sampling, systematic measurement error or a changing target process.

The interval estimates a population mean, not the middle 95% of individuals. Under the normal model, an individual prediction interval needs additional variability because a new observation differs from the mean even if the mean were known. The usual confidence interval becomes narrower with n while the inherent spread of individual outcomes remains.

If observations come without replacement from a finite population, account for the design or justify the usual small-sampling-fraction approximation. If observations are clustered or serially dependent, an independent-data standard error can underestimate uncertainty. The number of records is not always the effective number of independent units.

Keep units throughout: x̄, s, SE, margin and interval endpoints all have the measurement’s units. Variance has squared units. Writing these explicitly often reveals an incorrect denominator or accidental use of s² where s belongs.

Apply

For the practice interval, identify n, x̄, s, df and the supplied t critical value in separate lines. Then write a contextual conclusion and a sentence explaining what would make the procedure unreliable. Avoid claiming that a confidence interval guarantees the timing of an individual delivery.

Describe the t reference rather than only computing df

Changed-case comparison. Compare a t reference with 4 degrees of freedom, a t reference with 40 degrees of freedom and a standard normal reference. Describe symmetry, relative tail weight and the limiting relationship as df grows. Which t reference needs a larger positive cutoff to leave 2.5% above it? Explain why none assigns positive probability to exactly zero.

Feedback

All three are continuous and symmetric about zero. Both t references have heavier tails than the standard normal; the df=4 reference has heavier tails than df=40. Consequently its 97.5th-percentile cutoff is larger. As df increases, the t family approaches the standard normal. These are reference-curve comparisons, not comparisons of the raw sample distributions. A point has zero area under any of these continuous densities, even at the peak. For the one-sample mean procedure, df=n−1 and the reference accounts for estimating unknown population spread; merely computing df does not describe that uncertainty or establish that the study conditions hold.

Continue

Next use the same estimated standard error to test a specified mean. The null difference goes in the numerator, and the alternative determines the tail.