Statistics
Confidence Intervals II
March 6, 2009
Outline
1. A 95% confidence interval for µ is
σ
x̄ ± 1.96 √
n
2. Conditions for inference: normal distribution of population variable, SRS, known σ.
3. If σ is not known, use s, the sample standard deviation, to estimate it.
x̄ − µ
√
σ/ n
has a standard normal distribution but
x̄ − µ
√
s/ n
has a t-distribution with n − 1 degrees of freedom.
4. Properties of the t-distribution
5. The confidence interval now becomes
s
x̄ ± t∗ √
n
where t∗ is the appropriate critical value for the t distribution, use Table C.
6. Bottom line:
(a) Use s instead of σ
(b) Use a bigger critical value to account for uncertainty in not knowing σ (much bigger
if sample is small)