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Where we are going: a graphic: Samples 1 2 Estimation (1 sample). Paired Means Ho: / CI Ho: / CI Ho: / CI Variances Ho: / CI Ho: / CI Proportions Ho: / CI Ho: / CI Categories Ho: Ho: Slopes Ho: / CI 2 or more Ho: / CI 1 One Sample: Estimation. Gather data population sample Make inferences and comparisons parameters , 2 , , , , etc . statistics 2 ˆ, etc . , ˆ, ˆ , ˆ, ˆ or y , S2 S, p, b, etc. 2 Estimation. The (1-) % Confidence Interval mean --is a bound, or interval which, in repeated sampling, will cover the true mean (1- )% of the time. Sort of like horseshoes, hand grenades, or fishing with a net. Formula: mean 2 S (1 - ) % CI y t ,(n 1) n 2 Where: is the probability that the interval will be incorrect. ‘t’ is like z except appropriate for small samples the subscripts for t tell how to look it up in the table (appendix table 2 on page 1093 or the web page). 3 Pulse rate example: There were 128 measurements on 15 second pulse rate the mean was 19.60 and the standard deviation was 2.24. mean t 2 ,(n 1) t.025,127 1.96 when .05 So, a 95% CI is: 19.60 ± 1.96(2.24/11.31371) = 19.60 ± 0.39 = 19.21 << 19.99 we are 95% certain that this interval covers the true mean. 4 Estimation. The (1-) % Confidence Interval variance --is a bound, or interval which, in repeated sampling, will cover the true variance (1- )% of the time. Same as for the mean. Formula: variance (1 - )% CI 2 (n - 1)S 2 ,(n 1) 2 2 2 (n - 1)S 2 1 ,(n 1) 2 Where: is the probability that the interval will be incorrect. ‘2’ is a continuous distribution based on t2. See appendix table 7 on page 1100 (or the web page). 5 Pulse rate example: variance : n=128 mean=19.6 s2=5.0312 2 1 - ,(n 1 ) 2 2 ,( n 1 ) 2 2.975 ,( 127 ) 91.57 when .05 2 .025,(127) 152.21 when .05 (127)5.0132 (127)5.0132 2 152 .21 91.57 So, a 95% CI is: or 4.18 2 6.95 2.05 2.64 we are 95% certain that this interval covers the true variance 6 or standard deviation. Estimation. The (1-) % Confidence Interval proportion --is a bound, or interval which, in repeated sampling, will cover the true proportion (1- ) % of the time. Again, just like the mean (large samples only). Formula: proportion p(1 - p) (1 - )%CI p t ,(n 1) n 2 Where: is the probability that the interval will be incorrect. ‘t’ = z since the sample size is large. 7 Pulse rate example: proportion : we might wonder what proportion of people have a pulse rate 18 or less. P = 36/128 = .28 t 2 ,(n 1) t.025,127 1.96 when .05 So, a 95% CI is: .28 1.96 .28(.72) 128 .28 .08 .20 .36 we are 95% certain that this interval covers the true mean. The ‘.08’ is often called the ‘margin of error’. 8