MATH2209 Chap.5 Sampling, Estimation and Confidence Intervals
Sampling, Estimation and Confidence Intervals
Estimation connects a sample statistic to an unknown population parameter while acknowledging sampling variability. This chapter distinguishes probability and convenience sampling, separates bias from precision, and moves from point estimates to confidence intervals.
Repeated-interval diagrams clarify what the confidence level means, while worked calculations show how variability, critical value and the square root of sample size control the margin of error.
What this chapter covers
- 01
Probability sampling methods
- 02
Selection bias and representativeness
- 03
Central limit theorem
- 04
Point and interval estimates
- 05
Standard error and precision
- 06
Confidence interval interpretation
Worked example · free
Estimate a population mean height
- 1SE = 4.8 / sqrt(64) = 0.60 cm.
- 1Margin = 1.96 x 0.60 = 1.176 cm.
- 1Lower endpoint = 18.4 - 1.176 = 17.224 cm.
- 1Upper endpoint = 18.4 + 1.176 = 19.576 cm.
- 1Report 17.22 to 19.58 cm with a procedure-based interpretation.
Key terms
- Point Estimate
- A single sample value used to estimate a population parameter.
- Standard Error
- The standard deviation of an estimator across repeated samples.
- Confidence Level
- The long-run proportion of intervals from a method that capture the target parameter.
Sampling, Estimation and Confidence Intervals FAQ
What makes an interval narrower?
With other inputs fixed, a larger sample reduces standard error and narrows the interval. Lower confidence or lower variability also narrows it, but each change has a different interpretation. Precision should never be improved by silently changing the scientific target.
Why is confidence not a probability about this parameter?
In the frequentist procedure taught here, the population parameter is fixed while samples and intervals vary. The confidence level describes the method's long-run capture performance. The observed endpoints either contain the parameter or they do not.
Can a large sample remove selection bias?
No. Larger samples reduce random sampling variability under the design, but they do not restore groups that had no chance to enter. A very large convenience sample can be highly precise and still systematically miss the target population.
What should I check before interpreting an interval?
Name the population parameter, confirm that the point estimate sits at the midpoint and verify that both endpoints retain the measurement unit. Identify the sampling method and decide whether selection, nonresponse or measurement could create systematic error that the margin of error does not capture. Check that the standard error matches the setting and that the critical value belongs to the stated confidence level.
Interpret confidence as the long-run performance of the interval procedure, not as probability moving around a fixed parameter and not as a range containing most individuals. Then compare both endpoints with scientifically meaningful values. A narrow interval is useful only when the sample and measurement process connect the estimate to the intended population.
Assessment move
Practise drawing a population-to-sample-to-interval flow and label what varies at each stage. Identify the population parameter as fixed, the sample statistic as variable across samples and the confidence interval as a procedure built from that statistic. This diagram makes it easier to explain why one observed interval does not assign a probability to a fixed mean.
For calculation practice, use a fixed dataset to find the point estimate, standard error, critical value, margin of error and endpoints on separate lines. Attach units to the estimate, margin and interval. Check that the point estimate lies at the midpoint and that the lower endpoint is smaller. Recompute the interval after changing one input at a time.
Increase n while holding confidence and variability fixed, then explain the inverse square-root effect on standard error. Raise the confidence level and observe the wider critical multiplier. Increase variability and connect it with wider sampling uncertainty. These comparisons should be justified through the formula, not memorised as slogans. Pair precision work with sampling-design questions.
Decide whether the selection method is simple random, systematic, stratified, cluster or convenience, and distinguish random sampling from random treatment allocation. Ask whether the sampling frame omits part of the target population and whether nonresponse could be systematic. A small standard error cannot repair those biases. Write interval interpretations with the parameter, population and units.
Use long-run coverage language and avoid saying that 95% of individuals lie between the endpoints or that the fixed parameter has a 95% chance of being there. Compare the interval with scientifically important values rather than relying only on whether it includes a null value. End each session by reviewing one wide interval and proposing a defensible improvement.
More random observations may improve precision, a better sampling frame may improve representativeness, and a better measurement process may reduce variability; name which problem the change actually addresses.
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