MATH2089 Chap.10 Sampling and Confidence Intervals
Sampling and Confidence Intervals
Sampling distributions describe how statistics vary across repeated samples and provide the standard errors used in inference. The central limit theorem supports approximate normal behaviour for sample means under suitable conditions.
This chapter builds z, t and proportion intervals from an estimate, critical value and standard error, explains frequentist coverage, compares mean response with prediction targets and reverses the margin formula for sample size planning. A statistic changes from sample to sample even when the population parameter is fixed. Its sampling distribution describes that repeated-sample variation.
The standard error is the standard deviation of that distribution or an estimate of it. It is not the sample standard deviation itself: one describes variation among observations, while the other describes variation of an estimator.
The central limit theorem supports an approximate normal sampling distribution for a sample mean when observations are independent and the sample is sufficiently informative relative to population shape. It does not remove bias from a poor sampling design. Severe skew or influential observations matter more for a small sample, and independence must come from the way data were collected.
A confidence interval has the common form estimate plus or minus a critical value times standard error. The appropriate critical value depends on the parameter, information about variability and sampling model. A t interval for an unknown population mean uses an estimated standard error and degrees of freedom. A proportion interval needs enough expected successes and failures for the chosen approximation.
Print the estimate, standard error, critical value and margin separately before calculating endpoints. Confidence describes the long-run performance of the method. Across repeated samples, a 95% procedure is designed to cover the fixed parameter about 95% of the time under its conditions. After one interval is observed, the parameter is not randomly moving inside it.
Interpret the realised range as values compatible with the data and procedure, and name the population parameter and unit. Interval width follows useful direction checks. Higher confidence increases the critical value and widens the interval. Larger samples reduce standard error at a square-root rate, so halving a margin generally requires about four times the sample size. Greater variability widens the interval.
These relationships can reject an impossible calculator output quickly. Sample-size planning reverses the margin formula. Specify a target margin, confidence level and planning value for variability, then round upward. Prediction for one new observation is a different target from estimating a population mean and carries extra individual variation.
Finish with the design and distribution conditions beside the interval, because a narrow interval from a biased sample is still precisely biased.
What this chapter covers
- 01
Sampling distributions
- 02
Central limit theorem
- 03
Standard error
- 04
Known variance z intervals
- 05
Unknown variance t intervals
- 06
Proportion intervals and sample size
Worked example · free
Construct a t interval
- +1Standard error is 5 divided by the square root of 25, which equals 1.
- +1The margin is 2.064.
- +1The interval is 42 plus or minus 2.064.
- +1The endpoints are 39.936 and 44.064, interpreted under the sampling conditions.
Key terms
- Sampling distribution
- The distribution of a statistic across repeated samples under one design.
- Standard error
- The standard deviation of a statistic's sampling distribution.
- Coverage probability
- The long run proportion of intervals from a procedure that contain the fixed parameter.
Sampling and Confidence Intervals FAQ
What does 95% confidence mean?
It describes the long run coverage of the interval procedure across repeated samples. It is not a 95% probability statement about a fixed parameter after one interval is observed.
Why does larger n narrow an interval?
Standard error usually falls at a square root rate. Multiplying sample size by four therefore approximately halves the margin when confidence and variability are unchanged.
Why use a t critical value?
When population standard deviation is unknown, estimating it from the sample adds uncertainty. The t distribution accounts for that through heavier tails and degrees of freedom.
Exam move
Choose the interval type before calculating and state its conditions. Rework the example at a higher confidence level and a larger sample size, predicting the width direction before computing. Practise the repeated sampling interpretation until it can be stated without assigning probability to the fixed parameter. Write estimate, standard error, critical value and margin on separate lines for every interval.
Compare a mean-response interval with a prediction interval at the same target and explain the extra source of width. Reverse the margin formula for a sample-size calculation, round upward and state the planning value used for variability or proportion. Add one biased-sampling example to remind yourself that a narrow interval does not repair the design.
Finish each conclusion with the population parameter, unit, plausible range and conditions that support the procedure.
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