When σ is unknown, a confidence interval for a population mean uses the t distribution: x̄ ± t* · s/√n. For paired data, first find each pair's difference and then build the same interval for the mean difference μd. Enter data or summary statistics, check the conditions, and read the interval in context.
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Here the population is known: μ = 100 and σ = 15, normal. Each line is one t interval from a new random sample. Intervals that miss μ are red. The confidence level comes from the selector above.
We almost never know σ, so we estimate it with s. That adds extra variability, so we use a t distribution: symmetric and centered at 0 like z, but with heavier tails. The t* critical values are bigger than z*.
Each sample size gives a different t curve. As df grows, the t curve gets closer to the standard normal curve, and t* gets closer to z*.
When each subject is measured twice (or subjects are matched in pairs), subtract within each pair. Then treat the differences as one sample: x̄d, sd, and n = number of pairs.
Random: random sample or random assignment. 10%: n ≤ 10% of the population when sampling without replacement. Normal/Large: population normal, or n ≥ 30, or a graph of the data shows no strong skew or outliers.
"We are 95% confident that the interval from 6.2 to 7.4 hours captures the true mean sleep time of all students at this school." Name the parameter and the context.
95% confidence means that if we took many random samples and built an interval from each, about 95% of those intervals would capture μ. It is not the probability that this one interval contains μ.
What happens to the margin of error, ME = t* · s/√n, when one thing changes and everything else stays the same.
| Change | Effect on t* or SE | Margin of error | Interval |
|---|---|---|---|
| Increase the confidence level (95% → 99%) | t* gets larger | Larger | Wider |
| Increase the sample size n | s/√n smaller, and t* a bit smaller | Smaller | Narrower |
| Quadruple n | s/√n is cut in half | About half | About half as wide |
| More variable data (larger s) | s/√n larger | Larger | Wider |
| Different sample, same n | x̄ and s change | Changes a little | Moves and changes width |
| Procedure | Parameter | Interval | Calculator |
|---|---|---|---|
| One-sample t interval | μ, the population mean | x̄ ± t* · s/√n, df = n − 1 | STAT → TESTS → TInterval |
| Paired t interval | μd, the population mean difference | x̄d ± t* · sd/√n, df = n − 1 (n = pairs) | TInterval on the list of differences |
AP-style multiple choice. Pick an answer to see the explanation.
The vocabulary for topic 4.2. Read the list, flip through flashcards, or quiz yourself. For every term in the unit, see the Unit 4 terminology guide.