Population Means (Summary)
- When the population is normal and/or the sample is large, a confidence interval for unknown population mean μ (mu) when σ (sigma) is known is:
where z* is 1.645 for 90% confidence, 1.96 for 95% confidence, and 2.576 for 99% confidence.
- There is a trade-off between the level of confidence and the precision of the interval estimation. For a given sample size, the price we have to pay for more precision is sacrificing level of confidence.
- The general form of confidence intervals is an estimate +/- the margin of error (m). In this case, the estimate = x-bar and
The confidence interval is therefore centered at the estimate and its width is exactly 2m.
- For a given level of confidence, the width of the interval depends on the sample size. We can therefore do a sample size calculation to figure out what sample size is needed in order to get a confidence interval with a desired margin of error m, and a certain level of confidence (assuming we have some flexibility with the sample size). To do the sample size calculation we use:
(and round up to the next integer). We estimate σ (sigma) when necessary.
- When σ (sigma) is unknown, we use the sample standard deviation, s, instead, but as a result we also need to use a different set of confidence multipliers (t*) associated with the t distribution. We will use software to calculate intervals in this case, however, the formula for confidence interval in this case is
- These new multipliers have the added complexity that they depend not only on the level of confidence, but also on the sample size. Software is therefore very useful for calculating confidence intervals in this case.
- For large values of n, the t* multipliers are not that different from the z* multipliers, and therefore using the interval formula:
for μ (mu) when σ (sigma) is unknown provides a pretty good approximation.