STAT1008 Chap.11 Confidence Intervals for Means and Proportions
Confidence Intervals for Means and Proportions
Confidence Intervals for Means and Proportions builds intervals of the form point estimate plus or minus margin of error. It covers the two aspects of judging an estimator, bias and sampling error, the z-interval for a mean when sigma is known, the t-interval, whose degrees of freedom are one fewer than the sample size, when it is not, and the interval for a proportion.
It insists on the course’s interpretation wording, explains undercoverage, and plans sample sizes from a target margin of error. Assessment questions typically ask for an interval, its interpretation in the prescribed wording, a comment on whether a claimed value is plausible, the assumptions behind the method, and what happens to the width if the confidence level or sample size changes.
What this chapter covers
- 01
Point and interval estimates
- 02
Bias, sampling error and efficiency
- 03
The z-interval for a mean
- 04
Student’s t and degrees of freedom
- 05
The interval for a proportion
- 06
Interpreting a confidence interval
- 07
Sample size for a mean or a proportion
Worked example · free
A 95% confidence interval for flat-white volume
- 1Standard error = 12/√40 = 1.8974 mL.
- 1Margin of error = 1.96 × 1.8974 = 3.72 mL.
- 1Interval: 236 ± 3.72, from 232.28 to 239.72 mL.
- 2We are 95% confident the mean volume is between 232.28 and 239.72 mL; 240 mL lies above it, so the claim looks too high.
Key terms
- Margin of error
- The margin of error is the critical value multiplied by the standard error of the point estimate.
- Degrees of freedom
- Degrees of freedom count the observations free to vary once the sample mean is fixed, n minus 1 here.
- Confidence level
- The confidence level is the long-run share of intervals from the method that would contain the parameter.
- Student t distribution
- The Student t distribution is a symmetric, bell-shaped distribution with heavier tails than the normal, indexed by its degrees of freedom.
- Point estimate
- A point estimate is a single sample statistic used as the best guess of a population parameter.
Confidence Intervals for Means and Proportions FAQ
Does a 95% interval have a 95% chance of containing the mean?
Not in the course’s interpretation. Once calculated, an interval either contains the parameter or it does not. The 95% describes the method: over many repeated samples, about 95% of intervals built this way would capture the true mean.
When should I use t instead of z?
Use t when the population standard deviation is unknown and the sample standard deviation stands in for it, with n minus 1 degrees of freedom. Use z when sigma is given, or for a proportion interval.
How do I choose p when planning a survey for a proportion?
Use an estimate from a previous study if one exists. Otherwise use p equal to 0.5, which maximises p(1 minus p) and so gives the largest, safest sample size for the chosen margin of error.
What makes a confidence interval wider?
A higher confidence level raises the critical value, a larger standard deviation raises the standard error, and a smaller sample raises it as well. Any of these widens the interval. A larger sample is the main way to narrow an interval without losing confidence.
What does undercoverage mean for a t-interval?
Undercoverage happens when an interval labelled 95 per cent captures the true mean less often than that. It arises with small samples from heavily skewed populations, because the symmetric t distribution cannot account for the long tail.
Exam move
Memorise the critical values 1.645, 1.96 and 2.58 and the decision tree: mean or proportion, then sigma known or unknown. Practise writing the interpretation sentence until it is automatic, and for every interval add one line on the assumptions used. End each session with a sample-size calculation, rounding the answer up to a whole number.
Rehearse the interpretation sentence aloud and in writing for means and for proportions, changing only the parameter and units. Then practise three judgement questions: whether a claimed value lies inside the interval, what a higher confidence level would do, and whether the assumptions are met for the data described.
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