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Lec 02 - CI &amp; sSignificance

# Lec 02 - CI &amp; sSignificance - Confidence Intervals...

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Confidence Intervals Problem: Automatic filler deposits liquid in a container. Find a 95% confidence interval to estimate the current mean contents per container. Collect random sample: X i , i=1…n iid Compute sample average: X X n i i n = = 1 1 Compute sample standard deviation: ( 29 s x x n x nx n i i n i i n 2 2 1 2 2 1 1 1 = - - = - - = =

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Compute 95% Confidence Interval [ ] X t S n n ± - 1 2 , / / α S n / called standard error distribution of the sample average r v X N n . . ( , ) mean = std dev = μ σ since σ unknown, use sample estimate S and t distribution n-1 is degrees of freedom (as n gets large, t distribution approaches standard normal) t n-1, .025 is the point on the t distribution with n-1 degrees of freedom such that the area to the right is 0.025 2
Examples A sample of size 9 is randomly selected. The sample average is 10.4 and the sample standard deviation is 0.2. Compute a 90% confidence interval and a 99% confidence interval. 90% 10 4 1 86 0 2 9 10 4 124 1 86 8 05 CI t : [ . . . ] [ . . ] . ,. ± = ± = 99% 10 4 3 355 0 2 9 10 4 224 3 355 8 005 CI t : [ . . . ] [ . . ] . ,. ± = ± = 3

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Interpretation of the 99% Confidence Interval Repeat sampling 10 6 times & obtain CIs Each CI will have (slightly) different center point and width On average, 99% of the CIs include true mean μ . μ P(CI does not include true μ )=.01 “p value” 4
Confidence Intervals for Various Situations 95 % Confidence Interval mean, σ known, sample size n n z x σ 025 .

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