1. South Dakota School of Mines & Technology Estimation Industrial Engineering

2. Estimation Interval Estimates (s unknown) Industrial Engineering

3. Confidence Intervals (s unknown)
Suppose we do not know the true variance of the population, but we can estimate it with the sample variance.
For large samples (>30), replace s2 with s2 and compute confidence interval as before. x 1 2 - = å n s i

4. Confidence Intervals (s unknown)
For small samples we need to replace the standard normal, N(0,1) , with the t-distribution. Specifically, n 1 - = t s x m

5. Confidence Interval (s unknown)1 - = t s x m
tn-1,a/2
tn-1,a/2
a/2
1 - a
tn-1

6. Confidence Interval (s unknown)
tn-1,a/2
tn-1,a/2
a/2
1 - a
tn-1 ) ( 1 2 / , a m - = n t s x P

7. Confidence Interval (s unknown)1 2 / , a m - = n t s x P
Miracle 17b
occurs n x s t 2 / , 1 a -

8. Example
Suppose in our light bulb example, we wish to estimate an interval for the mean with 90% confidence. A sample of 25 bulbs yields a sample mean of 1,596 and a sample variance of 10,000. n x s t 2 / , 1 a - 25
100
711 . 1
596 ,

9. Example
Suppose in our light bulb example, we wish to estimate an interval for the mean with 90% confidence. A sample of 25 bulbs yields a sample mean of 1,596 and a sample variance of 10,000. n x s t 2 / , 1 a - 25
100
711 . 1
596 , 1,

10. Example
Note that lack of knowledge of s gives a slightly bigger confidence interval (we know less, therefore we feel less confident about the same size interval).
1, , ,628.9
32.9
s known
s unknown
1, , ,630.2
34.2

11. A Final Word
Note that on the t-distribution chart, as n becomes larger,
hence, for larger samples (n > 30) we can replace the t-distribution with the standard normal. 2 / , 1 a z t n -