t distribution
When sigma is unknown, replace it by the sample SD s and use the t distribution instead of the normal. It has heavier tails, shaped by degrees of freedom n minus 1. The interval is mean plus or minus t* x s / square root of n.
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Analogy
The t distribution is a cautious normal curve, wider to admit that s is only an estimate.
Example
n = 16, mean 20, s = 4SE = 4 / 4 = 1df = 15, t* = 2.131 (95%)Interval: 20 plus or minus 2.131= 17.87 to 22.13
In practice
Standard for small samples, where the extra uncertainty in s really matters.
Practice
Flashcards
- When do you use t instead of z?When the population SD is unknown and you use the sample SD s.
- Degrees of freedom for a one-sample t interval?n minus 1.
Questions people ask
Why is the t distribution wider than the normal?
The sample SD varies from sample to sample, which adds uncertainty beyond the sample mean. The t distribution has heavier tails to reflect that. With few degrees of freedom the effect is large, and the critical value is noticeably above 1.96.
When does the t distribution become the same as the normal?
As degrees of freedom grow, the estimate s becomes very reliable, and the t curve approaches the normal curve. By about 30 degrees of freedom they are close, and with infinite degrees of freedom they are identical, so t* becomes 1.96.