Confidence Intervals for Means and Proportions
Turn the sampling distribution around to produce an interval and a stated coverage. The t-distribution replaces the normal when the population standard deviation is estimated.
What a learner can do afterwards
- Construct an interval for a mean with the t-distribution
- Construct an interval for a proportion and check the conditions
- State what the confidence level does and does not say about one interval
1 · Read
A confidence interval centres on the sample mean and stretches by a multiplier times the standard error. The standard error is sigma over the square root of n, so bigger samples pull the endpoints tighter. With a multiplier near 2 for 95 percent, a sample of 25 with spread 10 gives a margin of 4.
In practice sigma is rarely known, so the sample spread s takes its place and extra uncertainty enters. The t distribution with n minus 1 degrees of freedom accounts for that substitution with heavier tails. The rule is simple: sigma known or huge sample, use z; sigma estimated from a modest sample, use t. With 16 observations that means t with 15 degrees of freedom.
Intervals for a proportion centre on the sample share p-hat, with spread built from p-hat itself. In 200 trials with 60 successes, p-hat is 0.3. The normal approximation needs enough of both outcomes: at least 10 successes and 10 failures. Polls quoting plus or minus three points are reporting exactly this construction.
The confidence level describes the method, not any single finished interval. If the whole recipe ran again and again, about 95 percent of the resulting intervals would trap the true mean. One computed interval either covers the mean or it does not, so never say there is a 95 percent chance the truth sits inside it.
Centre on the estimate, stretch by the standard error, use t when sigma is estimated, and read the level as a long run rate.
More to read
8.1 A Single Population Mean using the Normal Distribution
When the population spread is known or the sample is large, a mean interval centres on the sample mean and stretches by a normal multiplier times the standard error. The standard error is sigma over the square root of n, so bigger samples pull the endpoints tighter. A 95 percent interval uses a multiplier near 2, a handy approximation of 1.96.
The confidence level describes the method, not any single interval. If the whole sampling and interval recipe ran again and again, about 95 percent of the resulting intervals would trap the true mean. One computed interval either covers the mean or it does not; the 95 percent belongs to the long run procedure.
8.2 A Single Population Mean using the Student t Distribution
In practice sigma is rarely known, so the sample standard deviation s takes its place and extra uncertainty enters. The t distribution with n minus 1 degrees of freedom accounts for that substitution with heavier tails than the normal curve. Small samples pay a visible price in wider intervals.
As n grows, the t curve melts into the normal curve and the distinction stops mattering. The working rule is simple: sigma known or huge sample, use z; sigma estimated from a modest sample, use t with n minus 1 degrees of freedom. Mixing them up is one of the most common interval errors.
8.3 A Population Proportion
Intervals for a proportion centre on the sample proportion p-hat with a spread built from p-hat itself. The standard error is the square root of p-hat times (1 minus p-hat) over n, and the normal multiplier stretches it into a margin of error. Polls report exactly this construction when they quote plus or minus three points.
The normal approximation needs enough of both outcomes: at least 10 successes and 10 failures is the usual gate. Near zero or one, or with tiny samples, the approximation wobbles and exact methods take over. Always check the counts before quoting the interval.
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