1. 95% Confidence Interval μ
We are trying to estimate an unknown parameter, like the μ height of all Stat students.
some unknown, fixed μ
Just for a second, let’s pretend that we know what the value of μ is.

2. 95% Confidence Interval μ
the CLT tells us that when we repeatedly sample from a population
the distribution of x will be…
some unknown, fixed μ
NORMAL …provided that our sample size is large enough

3. 95% Confidence Interval μ So if we take a sample and calculate the x…
So if we take a sample and calculate the x…
some unknown, fixed μ
Or if we’re unlucky, it will be really far away from the true μ…
Or maybe it will be a little lower than μ…
Maybe we get lucky and it will be exactly the same as μ…
Or it could be here …
Probably it will be a little bit off from μ…
!!! UNUSUAL !!!

4. 95% of our x’s will be within 2 st.devs of the true μ
95% Confidence Interval
We’re not going to get unusually large or small values of x very often…
μ (σ/√n)
Remember 68/95/99.7?
some unknown, fixed μ
95% of our x’s will be within 2 st.devs of the true μ
μ (σ/√n)

5. 95% Confidence Interval μ The middle 95% of any normal distribution is
μ (σ/√n)
The middle 95% of any normal distribution is
contained within 1.96 st.devs. of μ μ
 Middle 95% 
μ (σ/√n)

6. …so

7. 95% Confidence Interval μ When we get a sample and calculate x…
When we get a sample and calculate x…
μ (σ/√n)
Like me!!!
some fixed average: μ
Or me!!!
μ (σ/√n)
We don’t have any way to know whether it’s one of the “GOOD” 95%

8. 95% Confidence Interval μ Or if it’s one of the “BAD” 5%
Like me 
μ (σ/√n)
Or if it’s one of the
“BAD” 5%
some fixed average: μ
μ (σ/√n)
Or me 

9. if 95% of all x-bars will be within 2 stdevs of µ…
95% Confidence Interval
if 95% of all x-bars will be within 2 stdevs of µ…
…then 95% of the time, the true µ will be within 2 stdevs of x-bar!!!

10. 95% Confidence Interval μ Did the interval “capture” the true mean?
Did the interval “capture” the true mean?
μ (σ/√n)
some fixed average: μ
μ (σ/√n)
Take this x, for example.
We’ll build a 95% interval around it…

11. …this x-bar and it’s interval?
95% Confidence Interval
…this x-bar and it’s interval?
μ (σ/√n)
some fixed average: μ
μ (σ/√n)

12. …this x-bar and it’s interval?
95% Confidence Interval
…this x-bar and it’s interval?
μ (σ/√n)
some fixed average: μ
μ (σ/√n)

13. 95% Confidence Interval μ μ + 1.96 (σ/√n) some fixed average:
μ (σ/√n)
some fixed average: μ
μ (σ/√n)

14. 4/19/2019
Lecture 23

15. a FATHOM Demo?
here
a Demo on your TI83?