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The Practice of Statistics Third Edition Chapter 11: Inference for Distributions Copyright © 2008 by W. H. Freeman & Company Daniel S. Yates

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Same as chapter 10 for inference for a mean of a population , except more realistic. Both population parameters and are unknown. is estimated by the sample standard deviation s. The standard deviation of the sampling mean is estimated by. is called the standard error of the sample mean,

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The t distribution When is used the statistic that results is called the t statistic. Very similar to z statistic we have been using.

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t distribution Similar in shape to standard normal curve. Symmetric about zero. Spread of t distribution. Greater than that of normal distribution. As degrees of freedom increase the t, k; k is number of degrees of freedom, approaches N(0,1) s estimates more accurately as sample size increases.

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t distribution Upper-tail probability t statistic Upper – Tail Probability df.05.025.02.01 16.31412.7115.8931.82 101.8122.2282.3592.764 201.7252.0862.1972.528 301.6972.0422.1472.457 ::::: 1001.6601.9842.0812.364 10001.6461.9622.0562.330 z * ( 1.6451.962.0542.326 90%95%96%98% Confidence level C Table B – AP Stats. Formula sheet Table C – AP Stat.s Book

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Problem. 11.10 Level of phosphate in blood. Tend to vary normally over time. Following is data for a patient on six visits; {5.6, 5.1, 4.6,4.8, 5.7, 6.4}. Construct 90% CI. x-bar = 5.37, s =.67, n = 6, df = 5, t = 2.015 5.37± 2.015(.67/√6) (4.82,5.91) We are 90% confident that the mean level of phosphate of blood in the patients blood is between 4.82 and 5.91 mg/dl

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Problem 11.12 The yield in pounds of two varieties of tomatoes are compared. Each variety of tomatoes is grown on one half of 10 plots of land. The 10 differences (variety A – variety B) give x-bar = 0.34 and s = 0.83. Is there convincing evidence that variety A has the higher yield? a – b Ho: = 0, no difference in yield Ha: > 0, variety A has larger yield df = 9t = 0.34 -0/ (0.83/√10) = 1.295 p( t > 1.29) = 0.114 There is insufficient evidence at 0.05 level to reject the null hypothesis that the yields of the two varieties of tomatoes are the same. 11.4% of the all the samples of size 10 that could have been taken would give a result that is as extreme as this if the true mean difference is zero.

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Comparing two Means Two – Sample Problems Compare the responses to two treatments or to compare the characteristics of two populations. Separate sample from each treatment or population. No matching of units in the two samples. The two samples can be of different sizes. Assumptions for comparing two means We have two SRS’s from two distinct populations. the samples are independent Both populations are normally distributed. The means and STD. of the populations are unknown.

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There are four unknown parameters ParameterStatistic X-bar X-bar ss ss We may want to compare the two population means. 1)Confidence interval: – 2)Hypothesis test: H o :

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Two sample t procedures Confidence interval Draw SRS of size n 1 from a normal population with unknown mean 1 Draw SRS of size n 2 from a normal population with unknown mean 2 (x-bar 1 – x-bar 2 ) ± t*√s 1 2 /n 1 + s 2 2 /n 2 use df = smaller of (n 1 -1) or (n 2 -1); TI-83,84 will calculate more precise degrees of freedom. Hypothesis test test: H : t = (x-bar 1 - x-bar 2 ) – ( 1 2 becomes t = (x-bar 1 - x-bar 2 ) √ s 1 2 /n 1 + s 2 2 /n 2 √ s 1 2 /n 1 + s 2 2 /n 2

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General considerations when answering inference questions 1) Four important questions. Is the question a confidence interval or hypothesis test? Is the question regarding one sample or two? Matched pairs? Does the question involve means, x-bar, or proportions p-hat? Should you use z statistic or t statistic? ZAPTAX TLRTLM AWOAWE BAPBAA LYOLYN ESRES T I O N

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2) Follow four step procedure for Hypothesis test or Confidence interval.. H A M C State the hypothesis – null and alternate. Identify type of test and assumptions. Do the math; show formula and calculation. State the conclusion in context. IIndependence of samples RRandom selection; SRS OCheck outliers NIs data normally distributed SSample size; np>10, n(1-p)>10 10n<pop. Size, large sample Must do these three for a confidence interval

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