STAT 400 Chap.9 Confidence Intervals and Sample Size
Confidence Intervals and Sample Size
Sampling distributions come first
An interval estimate needs to know how a statistic varies from sample to sample. For a normal population three results do the work. The standardised sample mean with known σ is standard normal. The scaled sample variance (n − 1)S²/σ² follows a chi-square distribution with n − 1 degrees of freedom.
And replacing σ with S in the standardised mean gives Student's t distribution with n − 1 degrees of freedom. The course introduces the chi-square and t distributions in their own right, including reading their tables, before building any interval.
Degrees of freedom in plain terms
The deviations of the observations from the sample mean always sum to zero, so once n − 1 of them are known the last is fixed.
That is the sense in which estimating the mean costs one degree of freedom, and it is why the sample variance divides by n − 1. A t distribution with few degrees of freedom has heavier tails than the normal, so its critical values are larger; as the degrees of freedom grow, it approaches the standard normal.
Intervals for a mean, a proportion and a variance
Every interval in this chapter has the shape estimate plus or minus a critical value times a standard error, except the variance interval, which divides by two different chi-square values because the chi-square distribution is not symmetric.
For a mean, use z when σ is known and t with n − 1 degrees of freedom when it is estimated. For a proportion, the large-sample interval uses the sample proportion in its standard error.
The course also asks for one-sided confidence bounds, which put all of α in one tail and so use z_α rather than z_α/2.
Interpreting confidence
A 95% confidence level describes the method: across repeated samples, 95% of intervals built this way contain the parameter. A single computed interval either contains it or does not.
The correct sentence is that you are 95% confident the interval captures the population mean, not that the mean has a 95% probability of lying inside it.
Planning the sample size
Solving the margin-of-error formula for n turns an interval into a planning tool.
For a mean with known σ, n = (zσ/ε)²; for a proportion, n = z²p*(1 − p*)/ε², using p* = 0.5 when no prior estimate is available because it gives the largest, safest n. Always round up. Halving the margin requires four times the sample.
What this chapter covers
- 01
Chi-square and t distributions
- 02
Degrees of freedom
- 03
Intervals for a mean with σ known or unknown
- 04
Intervals and bounds for a proportion
- 05
Interval for a variance
- 06
Sample size for a target margin
Worked example · free
A t interval from summary statistics
- 2σ is unknown, so use t with 19 degrees of freedom; the upper 2.5% critical value is 2.093.
- 2Standard error = 0.58/√20 = 0.1297, so the margin is 2.093 × 0.1297 = 0.271.
- 2Interval: 3.42 ± 0.271 = (3.15, 3.69), read as the method capturing μ in 95% of samples.
Key terms
- Confidence Interval
- An interval computed from sample data by a method that captures the parameter in a stated proportion of repeated samples.
- Degrees of Freedom
- The number of independent pieces of information behind an estimate; n − 1 for the sample variance of n observations.
- Student's t Distribution
- The distribution of the standardised sample mean when σ is replaced by S, symmetric with heavier tails than the normal.
- Chi-Square Distribution
- The distribution of a sum of squared independent standard normals, used for inference about a normal variance.
- Margin of Error
- The half-width of a symmetric confidence interval, equal to a critical value times the standard error.
Confidence Intervals and Sample Size FAQ
When should I use t instead of z for a mean?
Use t whenever the population standard deviation is estimated from the same sample, which is the usual case. Use z only when σ is genuinely known from outside the data.
Why is the variance interval not centred on the sample variance?
The chi-square distribution is skewed to the right, so the two critical values sit at different distances from the centre, and dividing by them produces an interval that is longer above s² than below it.
Why use 0.5 for the proportion when planning a sample size?
The product p(1 − p) is largest at p = 0.5, so that choice gives the biggest required sample and guarantees the margin of error whatever the true proportion turns out to be.
Exam move
Write the general shape, estimate plus or minus critical value times standard error, then fill it in for each of the four intervals. Practise reading the t and chi-square tables with the upper-tail convention until the two chi-square values for a variance interval come out in the right order.
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