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95% Confidence Interval

0. We are trying to estimate an unknown parameter, like the μ height of all Stat students. 95% Confidence Interval. some unknown, fixed. μ. Just for a second, let’s pretend that we know what the value of μ is. 0. the CLT tells us that when we repeatedly sample from a population

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95% Confidence Interval

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  1. 0 • We are trying to estimate an unknown parameter, like the μ height of all Stat students. 95% Confidence Interval some unknown, fixed μ Just for a second, let’s pretend that we know what the value of μ is.

  2. 0 the CLT tells us that when we repeatedly sample from a population the distribution of x will be… 95% Confidence Interval some unknown, fixed μ NORMAL…provided that our sample size is large enough

  3. 0 So if we take a sample and calculate the x… 95% Confidence Interval 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. 0 We’re not going to get unusually large or small values of x very often… μ + 1.96 (σ/√n) 95% Confidence Interval 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)

  5. 0 μ + 1.96 (σ/√n) 95% Confidence Interval The middle 95% of any normal distribution is contained within 1.96 st.devs. of μ μ  Middle 95%  μ - 1.96 (σ/√n)

  6. …so 0

  7. 0 When we get a sample and calculate x… μ + 1.96 (σ/√n) 95% Confidence Interval 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%

  8. 0 Like me  μ + 1.96 (σ/√n) 95% Confidence Interval Or if it’s one of the “BAD” 5% some fixed average: μ μ - 1.96 (σ/√n) Or me 

  9. if 95% of all x-bars will be within 2 stdevs of µ… 95% Confidence Interval 0 …then 95% of the time, the true µ will be within 2 stdevs of x-bar!!!

  10. 0 Did the interval “capture” the true mean? μ + 1.96 (σ/√n) 95% Confidence Interval some fixed average: μ μ - 1.96 (σ/√n) Take this x, for example. We’ll build a 95% interval around it…

  11. 0 …this x-bar and it’s interval? μ + 1.96 (σ/√n) 95% Confidence Interval some fixed average: μ μ - 1.96 (σ/√n)

  12. 0 …this x-bar and it’s interval? μ + 1.96 (σ/√n) 95% Confidence Interval some fixed average: μ μ - 1.96 (σ/√n)

  13. 0 μ + 1.96 (σ/√n) 95% Confidence Interval some fixed average: μ μ - 1.96 (σ/√n)

  14. Lecture 23

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

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