chapter5 - Chapter 5 Inferences Regarding Population...

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Unformatted text preview: Chapter 5 Inferences Regarding Population Central Values Inferential Methods for Parameters Parameter : Numeric Description of a Population Statistic : Numeric Description of a Sample Statistical Inference : Use of observed statistics to make statements regarding parameters Estimation : Predicting the unknown parameter based on sample data. Can be either a single number ( point estimate) or a range ( interval estimate) Testing : Using sample data to see whether we can rule out specific values of an unknown parameter with a certain level of confidence Estimating with Confidence Goal: Estimate a population mean based on sample mean Unknown: Parameter ( ) Known: Approximate Sampling Distribution of Statistic n N Y , ~ 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 . 2 2 + - n Y n P Estimating with Confidence Although the parameter is unknown, its highly likely that our sample mean ( estimate ) will lie within 2 standard deviations (aka standard errors ) of the population mean ( parameter ) Margin of Error : Measure of the upper bound in sampling error with a fixed level (we will typically use 95%) of confidence. That will correspond to 2 standard errors: error of margin estimate : Interval Confidence 2 : ) Confidence (95% Error of Margin n Confidence Interval for a Mean Confidence Coefficient (1-29 Probability (based on repeated samples and construction of intervals) that a confidence interval will contain the true mean Common choices of 1- and resulting intervals: n z y n y n y n y 2 / : Confidence % 100 ) 1 ( 576 . 2 : Confidence 99% 960 . 1 : Confidence 95% 645 . 1 : Confidence 90% - 1- /2 1- /2 z_ /2 0.90 0.050 0.950 1.645 0.95 0.025 0.975 1.960 0.99 0.005 0.995 2.576 Standard Normal Distribution 2 / z 2 / z- 1- 2 2 Normal Distribution n z 2 / + n z 2 /- 1- 2 2 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 4.33 0.0081 49 5.37 0.0085 101 5.88 0.0133 7 2.06 0.0145 330 4.03 0.0101 398 2.29 0.0166 33 0.79 0.0148 312 4.51 0.0153 468 4.63 0.0132 396 5.55 0.0187 299 2.60 0.0164 219 4.41 0.0160 293 3.99 0.0145 99 2.222....
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chapter5 - Chapter 5 Inferences Regarding Population...

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