ETF1100 Chap.5 Sampling and Confidence Intervals
Sampling and Confidence Intervals
Sampling and Confidence Intervals explains why a statistic changes across random samples and how standard error quantifies that fluctuation. The chapter builds intervals as an estimate plus a margin of error, distinguishes confidence from precision, gives the correct repeated-sampling interpretation, and separates sampling uncertainty from selection bias, measurement error and nonresponse.
What this chapter covers
- 01
Population parameters and sample estimates
- 02
Random sampling and generalisation
- 03
Sampling distributions
- 04
Standard errors
- 05
Confidence level and margin
- 06
Coverage interpretation and bias
Worked example · free
Construct a mean confidence interval
- 1Compute SE = 8.0 / sqrt(64) = 1.00 hour.
- 1Compute margin = 1.96 x 1.00 = 1.96 hours.
- 1Subtract the margin to obtain 29.54 hours.
- 1Add the margin to obtain 33.46 hours.
- 1Interpret the range for the population mean, not individual orders.
Key terms
- Population Parameter
- A fixed but usually unknown numerical feature of a population.
- Sample Estimate
- A statistic calculated from a sample to estimate a population parameter.
- Sampling Distribution
- The distribution of an estimator over repeated samples from the same process.
- Standard Error
- The standard deviation of an estimator's sampling distribution.
- Margin Of Error
- The critical multiplier times standard error, forming the half-width of an interval.
Sampling and Confidence Intervals FAQ
What does 95% confidence mean?
It describes a method that captures the true population parameter in about 95% of repeated random samples under its assumptions. For one realised interval, describe a range of plausible parameter values rather than assigning a 95% probability to the fixed parameter.
How can I make a confidence interval narrower?
A larger sample usually reduces standard error, while lower confidence uses a smaller critical multiplier. Less observed variability also narrows an interval. Reducing confidence trades coverage for precision, so it is not a free improvement.
Does a narrow interval remove sampling bias?
No. Width quantifies sampling fluctuation under the calculation. A convenience sample can produce a narrow interval while systematically excluding part of the population. Defend the sampling frame and selection method separately from the margin of error.
What quick check can catch an endpoint error?
Recover the midpoint and half-width from the reported endpoints. The midpoint should equal the point estimate and both endpoints should be equally distant from it. They must appear in increasing order and carry the parameter's unit. This check detects misplaced signs or asymmetric arithmetic without repeating the full calculation.
Exam move
Draw a fixed population parameter and several different sample estimates to make sampling variability concrete. Learn the interval architecture as estimate plus or minus critical value times standard error, then practise predicting how width changes when sample size, confidence or variability changes. Write one correct interpretation for a mean and one for a proportion, always naming the population parameter.
Cross out statements that claim a confidence percentage of individual observations or a probability that the fixed parameter lies inside one completed interval. For each problem, separate random sampling from convenience selection and list sources of bias that an interval does not address.
Compare the interval with a practical target: if the full range lies on one side, the decision is clearer; if it crosses the threshold, more evidence may be useful. Keep intermediate standard errors unrounded. On the exam, verify whether the question requests an endpoint, margin, width or full interval before entering a number.
Build an interval worksheet with labelled cells for estimate, sample size, variability, standard error, critical multiplier, margin and endpoints. Before calculating, predict whether each design change makes the interval wider, narrower or unchanged.
Then vary one input at a time and explain the result: quadrupling sample size roughly halves the standard error, increasing confidence widens the interval, and greater variability reduces precision. Keep selection bias in a separate box because no multiplier fixes a frame that excludes important units.
Practise reading the same interval against three business thresholds; the numerical interval stays fixed while the decision implication changes. For every interpretation, underline the parameter and represented population, circle the unit and reject any sentence about the percentage of individual observations between endpoints.
Compare two samples only with a method suited to their design; visual overlap of separate intervals is not a complete test of a difference. If data are paired, record the pairing rather than treating observations as unrelated. Use a rough magnitude check to catch an endpoint placed on the wrong side of the estimate.
Finally, explain when more data would be valuable: not simply when the interval is wide, but when plausible values imply meaningfully different actions. That connects precision to the decision instead of treating narrowness as an end in itself.
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