Lesson 29 of 1524
Confidence intervals
The percent names the method, not the chance that this finished interval contains the parameter.
Practice this chapterA confidence interval is a point estimate plus or minus a margin of error. In the long run, 95% of intervals made this way capture the parameter. After one interval is calculated, the parameter is either inside it or not.
Unknown σ uses a t critical value with n − 1 degrees of freedom. Raising the confidence level widens the interval. Raising n narrows it. A proportion interval in this text needs at least 5 successes and 5 failures.
Degrees of freedom for a one-sample t interval
df = n − 1
One less than the sample size.
Worked example
A sample has 12 observations and σ is unknown. What t distribution do you use?
- 1σ unknown means the spread was estimated from this same sample. The critical value then comes from a t distribution, not from the standard normal z.
- 2For a one-sample mean, the degrees of freedom are n − 1.
- 3n = 12, so df = 12 − 1 = 11.
- 4You use the t curve with 11 degrees of freedom. Its tails are heavier than the z curve, so the critical value is a little larger than the matching z value.
Result: t with 11 degrees of freedom
Why. You estimated the mean and you also estimated the standard deviation from the same 12 observations. That second estimate costs one degree of freedom, leaving 11. The t curve, not z, is the one that matches an estimated spread.
Do not say “there is a 95% chance that this interval contains μ.” The 95% is the success rate of the procedure.
Practice margin
This chapter
A fresh set from this chapter only. Choose 10 or 20. Multiple choice and fill-in, with no repeat inside the set.