Confidence interval for a mean
A confidence interval is a range likely to contain the true parameter. With known sigma, a 95 percent interval is the sample mean plus or minus 1.96 x sigma / square root of n.
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Analogy
A fisherman's net rather than a spear: you do not claim an exact spot, you cast a range likely to catch the truth.
Example
Mean 52, sigma 10, n = 25SE = 10 / 5 = 2Margin = 1.96 x 2 = 3.92Interval: 48.08 to 55.92
In practice
Reports give an average with an interval to show how precise the average is.
Practice
Flashcards
- z value for a 95% confidence interval?1.96.
Interview questions
- A poll of 400 voters finds 56% support a law. Calculate and interpret a 95% confidence interval.
- SE = 0.0248
- Margin = 1.96 x 0.0248 = 0.0486
- Interval 51.1% to 60.9%
- Repeated polls would capture the true share about 95% of the time
Questions people ask
What does 95 percent confidence actually mean?
If you repeated the sampling many times and built an interval each time, about 95 percent of those intervals would contain the true mean. It does not mean this one interval has a 95 percent chance, since the true mean is fixed and either inside or not.
How can I make a confidence interval narrower?
Increase the sample size, which shrinks the standard error, or accept lower confidence, such as 90 percent instead of 95. Reducing the natural variability of the measurements helps too. Narrow and highly confident together needs a larger sample, so the real trade-off is between cost, precision and certainty.