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chapter5 - Chapter 5 Inferences Regarding Population...

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Chapter 5 Inferences Regarding Population Central Values
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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
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Estimating with Confidence Goal: Estimate a population mean based on sample mean Unknown: Parameter ( μ ) Known: Approximate Sampling Distribution of Statistic n N Y σ μ , ~ 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 . 0 2 2 + - n Y n P σ μ σ μ
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Estimating with Confidence Although the parameter is unknown, it’s 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 σ
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Confidence Interval for a Mean μ Confidence Coefficient (1 - α 29 Probability (based on repeated samples and construction of intervals) that a confidence interval will contain the true mean μ Common choices of 1- α and resulting intervals: n z y n y n y n y σ α σ σ σ α 2 / : Confidence % 100 ) 1 ( 576 . 2 : Confidence 99% 960 . 1 : Confidence 95% 645 . 1 : Confidence 90% ± - ± ± ± 1- α α /2 1- α /2 z_ α /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
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Standard Normal Distribution 2 / α z 2 / α z - 1- α 2 α 2 α 0
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Normal Distribution n z σ μ α 2 / + n z σ μ α 2 / - 1- α 2 α 2 α μ
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Philadelphia Monthly Rainfall (1825-1869) Histogram 0 20 40 60 80 100 120 140 1 3 5 7 9 11 13 15 More Frequency 1 2 3 4 5 6 7 8 9 10 11 12 13 14 84 . 0 20 92 . 1 96 . 1 : %) 95 20, ( error of Margin 92 . 1 68 . 3 = = = = = C n σ μ
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4 Random Samples of Size n =20, 95% CI’s 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 2.22 0.0233 337 1.85 0.0191 16 3.87 0.0171 511 2.39 0.0149 181 1.84 0.0235 447 3.55 0.0193 484 2.83 0.0190 235 5.28 0.0172 364 2.05 0.0244 78 3.53 0.0213 316 4.56 0.0202 314 3.11 0.0190 392 7.59 0.0253 117 3.57 0.0224 318 3.44 0.0257 372 5.42 0.0260 477 7.16 0.0283 399 1.09 0.0227 517 3.62 0.0272 164 2.78 0.0272 434 2.07 0.0290 52 4.99 0.0240 249 2.16 0.0301 48 0.26 0.0281 229 4.05 0.0318 162 6.60 0.0261 445 4.79 0.0320 236 2.40 0.0284 223 4.54 0.0320 95 2.59 0.0296 13 1.11 0.0324 50 3.75 0.0319 279 2.76 0.0364 479 3.93 0.0296 479 3.93 0.0325 39 3.35 0.0325 520 5.44 0.0374 51 2.87 0.0303 370 4.11 0.0345 417 7.68 0.0333 245 1.60 0.0374 380 6.00 0.0311 348 2.17 0.0374 503 1.76 0.0359 183 2.63 0.0391 61 1.63 0.0324 89 5.40 0.0380 151 5.89 0.0361 41 3.49 0.0395 302 2.87 0.0339 Mean 3.39 3.88 3.86 3.15 Mean-me 2.55 3.04 3.02 2.31 Mean+me 4.23 4.72 4.70 3.99 84 . 0
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