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.

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

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 !!!

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 + 1.96 (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 - 1.96 (σ/√n)

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

…so

95% Confidence Interval μ When we get a sample and calculate x… When we get a sample and calculate x… μ + 1.96 (σ/√n) Like me!!! some fixed average: μ Or me!!! μ - 1.96 (σ/√n) We don’t have any way to know whether it’s one of the “GOOD” 95%

95% Confidence Interval μ Or if it’s one of the “BAD” 5% Like me  μ + 1.96 (σ/√n) Or if it’s one of the “BAD” 5% some fixed average: μ μ - 1.96 (σ/√n) Or me 

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!!!

95% Confidence Interval μ Did the interval “capture” the true mean? Did the interval “capture” the true mean? μ + 1.96 (σ/√n) some fixed average: μ μ - 1.96 (σ/√n) Take this x, for example. We’ll build a 95% interval around it…

…this x-bar and it’s interval? 95% Confidence Interval …this x-bar and it’s interval? μ + 1.96 (σ/√n) some fixed average: μ μ - 1.96 (σ/√n)

…this x-bar and it’s interval? 95% Confidence Interval …this x-bar and it’s interval? μ + 1.96 (σ/√n) some fixed average: μ μ - 1.96 (σ/√n)

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

4/19/2019 Lecture 23

a FATHOM Demo? here a Demo on your TI83?