COMM5000 Chap.4 Confidence Intervals and Sample Size
Confidence Intervals and Sample Size
Construct and interpret confidence intervals, explain the long-run procedure, and plan sample size by balancing confidence, precision, variability and collection quality. An interval combines a point estimate with a margin of error. The margin equals a critical value multiplied by the estimator's standard error, so interval width reflects both the chosen confidence level and sampling precision.
Higher confidence uses a larger critical value and produces a wider interval when other inputs remain fixed. Larger samples usually reduce standard error and narrow the interval according to a square-root relationship. Interpret the confidence level as a property of the repeated procedure: over many comparable samples, the stated proportion of intervals captures the fixed population parameter.
Do not assign that probability to the parameter after one interval is calculated. For proportions, check that the normal approximation is suitable; for means, match the procedure to what is known about variability and the sample conditions. Sample-size planning begins with a target margin of error and rounds the calculated requirement upward.
The formula addresses random sampling variation but not nonresponse, coverage or measurement bias. A narrow interval can therefore be precise and still misleading. In a business report, place the interval beside the decision threshold, explain whether the plausible range changes the action, and identify the most important uncertainty outside the formula.
What this chapter covers
- 01
Point estimate
- 02
Margin of error
- 03
Confidence procedure
- 04
Interval width
- 05
Sample-size planning
Worked example · free
Recommendation proportion interval
- +1The four marks used here are not an official university marking scheme. p-hat = 236/400 = 0.59.
- +1SE = sqrt(0.59 x 0.41/400) = 0.02459.
- +1Margin = 1.96 x 0.02459 = 0.04820.
- +1Interval = 0.5418 to 0.6382.
Key terms
- Margin of error
- The critical value multiplied by the estimator's standard error.
- Confidence level
- The long-run capture rate of a confidence-interval procedure.
- Precision
- The degree to which an estimator has small repeated-sample variation.
Confidence Intervals and Sample Size FAQ
What does 95% confidence mean?
Across repeated samples analysed with the same valid procedure, about 95% of the resulting intervals contain the fixed population parameter. It is not a probability statement about one completed interval.
Why round sample size upward?
The design sets a maximum margin of error. Rounding down produces less precision than required, so the calculated value must be taken to the next whole observation.
What should accompany a confidence interval in a report?
Name the population parameter, estimate, confidence level and sampling process, then state the interval in consistent units. Explain whether its plausible range lies above, below or across the business threshold that drives the decision. Add the largest uncertainty that the formula does not cover, such as nonresponse, frame exclusions or measurement error.
If the achieved sample differs from the plan, recompute the actual precision. This combination lets a reader audit both the numerical procedure and the design limits instead of treating narrow bounds as proof that every source of uncertainty is small.
Assessment move
Write interval calculations in a fixed audit order: estimate, standard error, critical value, margin of error, lower bound and upper bound. Retain unrounded intermediate values and round only the reported limits. Practise converting between proportions and percentages without mixing scales. After each interval, name the population parameter, sampling process and confidence procedure in one sentence.
Then write a second sentence identifying a bias threat that the margin of error does not represent. For planning questions, solve the margin formula for n, substitute the target confidence and precision, and round upward. Explain how confidence, variability and sample size move interval width before performing arithmetic.
When an interval crosses a business threshold, describe the range of supported actions instead of forcing a single conclusion. If the achieved sample differs from the plan, recompute actual precision. In assessment work, show enough calculation for a reviewer to reproduce both limits and verify the decision.
Keep calendar, recruitment and response assumptions in the methods section so the numerical precision is not detached from how the data were obtained. Create comparison cards for 90%, 95% and 99% procedures using the same estimate and standard error. Describe why the bounds widen without saying that the data changed. Repeat the exercise with four times the sample size and verify that the standard error halves.
For a planning calculation, write the inequality that the margin must satisfy, solve for n and take the ceiling. Record the expected response rate separately from the completed-sample requirement so recruitment targets are realistic. On the final page of a report, place the interval near the relevant decision threshold and explain whether all, some or none of the plausible range supports the proposed action.
That presentation gives a decision-maker more information than a confidence label alone.
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