chapter6 - Chapter 6 Introduction to Statistical Inference...

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Unformatted text preview: Chapter 6 Introduction to Statistical Inference Introduction Goal: Make statements regarding a population (or state of nature) based on a sample of measurements Probability statements used to substantiate claims Example: Clinical Trial for Pravachol (5-year follow-up) Of 3302 subjects receiving Pravachol, 174 had heart incidences Of 3293 subjects receiving placebo, 248 had heart incidences 11363 in chance one ely Approximat .000088 : effective not if better much this do would Pravachol y that Probabilit %) 53 . 7 ( 0753 . 3293 248 %) 27 . 5 ( 0527 . 3302 174 placebo ^ Pravachol ^ = = = = p p Estimating with Confidence Goal: Estimate a population mean (proportion) based on sample mean (proportion) Unknown: Parameter ( , p ) Known: Approximate Sampling Distribution of Statistic - n p p p N p n N X ) 1 ( , ~ , ~ ^ Recall: For a random variable that is normally distributed, the probability that it will fall within 2 standard deviations of mean is approximately 0.95 95 . ) 1 ( 2 ) 1 ( 2 95 . 2 2 ^ - + -- + - n p p p p n p p p P n X n P Estimating with Confidence Although the parameter is unknown, its highly likely that our sample mean or proportion ( estimate ) will lie within 2 standard deviations (aka standard errors ) of the population mean or proportion ( parameter ) Margin of Error : Measure of the upper bound in sampling error with a fixed level (we will use 95%) of confidence. That will correspond to 2 standard errors: error of margin estimate : Interval Confidence ) 1 ( 2 : ) Confidence (95% Error of Margin : Proportion 2 : ) Confidence (95% Error of Margin : Mean - n p p n Confidence Interval for a Mean Confidence Coefficient ( C ): Probability (based on repeated samples and construction of intervals) that a confidence interval will contain the true mean Common choices of C and resulting intervals: n z x C n x n x n x * : Confidence % 576 . 2 : Confidence 99% 960 . 1 : Confidence 95% 645 . 1 : Confidence 90% C z* 90% 1.645 95% 1.960 99% 2.576 Normal Distribution n z * + n z *- C 2 1 C- 2 1 C- Standard Normal Distribution * z * z- C 2 1 C- 2 1 C- Philadelphia Monthly Rainfall (1825-1869) Histogram 20 40 60 80 100 120 140 1 3 5 7 9 1 1 1 3 1 5 M o r e Frequency 1 2 3 4 5 6 7 8 9 10 11 12 13 14 84 . 20 92 . 1 96 . 1 : %) 95 20, ( error of Margin 92 . 1 68 . 3 = = = = = C n 4 Random Samples of Size n =20, 95% CIs Sample 1 Sample 2 Sample 3 Sample 4 Month Rain Ran# Month Rain Ran# Month Rain Ran# Month Rain Ran# 156 2.56 0.0028 349 2.33 0.0007 185 2.69 0.0005 171 1.50 0.0011 51 2.87 0.0050 149 4.86 0.0013 527 5.28 0.0029 175 2.52 0.0048 176 4.64 0.0052 227 4.15 0.0054 114 3.99 0.0048 130 1.22 0.0085 364 2.05 0.0082 336 5.17 0.0073 312 4.51 0.0084 167 3.35 0.0094 271 2.76 0.0142 124 4.334....
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This note was uploaded on 06/04/2011 for the course STA 2023 taught by Professor Ripol during the Fall '08 term at University of Florida.

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chapter6 - Chapter 6 Introduction to Statistical Inference...

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