Introductory Applied Statistics for the Life Sciences
Introductory Applied Statistics for the Life Sciences STAT 371
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Mrs. Triston Collier
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This 9 page Class Notes was uploaded by Mrs. Triston Collier on Thursday September 17, 2015. The Class Notes belongs to STAT 371 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 35 views. For similar materials see /class/205079/stat-371-university-of-wisconsin-madison in Statistics at University of Wisconsin - Madison.
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Date Created: 09/17/15
Review le1Xn have mean u and SD 0 EX u no matter what SDX am if the X s are independent If X1Xn are iid normalmeanu SD0 X N normalmean M SD an If X1Xn are iid with mean u and SD 0 and the sample size n is large X N normalmean M SD an A discrepancy Caution Sometimes the order in which the book covers material is a bit odd The authors would probably think I m odd But sometimes it is just wrong Or perhaps they are making some simplifications to ease learning A case in point Let X1 Xn be random draws from a population with mean p and SD 0 and X the sample average Book Karl a population SD population SD 5 SD of the data our estimate of a a SD of the sampling distribution of X SDX aka SEX s Standard error of the mean our estimate of SEX Confidence intervals Suppose we measure the log1o Cytokine response in 100 male mice of a certain strain and find that the sample average Sr is 352 and sample SD 3 is 161 Our estimate of the SE of the sample mean is 161x100 0161 A 95 confidence interval for the population mean u is 352 i 2 x 016 352 i 032 320 384 What does this mean What is the chance that 320 384 contains u Suppose that X1 Xn are iid normalmeanp SD0 Suppose that we actually know 0 Then X N normalmeanu SDan where 0 is known but u is not How close is X to u Pr Ifya g 196 95 1960 7 1960 P ltX lt 95 o r w 7 1960 7 1960 P X lt ltX 95 o r w w What is a confidence interval A 95 confidence interval is an interval calculated from the data that in advance has a 95 chance of covering the population parameter In advance Xi 1960n has a 95 chance of covering u Thus it is called a 95 confidence interval for u Note that after the data is gathered for instance n100 X 352 s 161 the interval becomes fixed Xi1960f 352 i 032 We can t say that there s a 95 chance that u is in the interval 352 i 032 It either is or it isn t we just don t know 500 confidence intervals for u 6 known I I I I I I 1 a I E I I I 1 1 El I E I r L I I I z I 1 I I I L I I I I I I I 1 I I I 1 I I E r L I I I a a E E I L I 2 l l El I E E I E r r J L EE 5 E E a E E E 5 I 139 E E E E 5 5 E E E E E a 1 E E a a 3 E E39 E E i 3 E E E E E E f E E E E E I E E E E E E E a E E E E E E E E E E j 3 f E i 39 I I I39 I I I I I Longer and shorter intervals If we use 164 in place of 196 we get shorter intervals with lower confidence lX ul s P lt164 90 Ince r0 Xi 164an is a 90 confidence interval for u If we use 258 in place of 196 we get longer intervals with higher confidence l Ml Since Pr lt 258 99 a X i 25mm is a 99 confidence interval for u What is a confidence interval A 95 confidence interval is obtained from a procedure for producing an interval based on data that 95 of the time will produce an interval covering the population parameter In advance there s a 95 chance that the interval will cover the population parameter After the data has been collected the confidence interval either contains the parameter or it doesn t Thus we talk about confidence rather than probability But we don t know the SD Use of X i 196 am as a 95 confidence interval for u requires knowledge of 0 That the above is a 95 confidence interval for u is a result of the following X M 0W N normal01 What if we don t know a We plug in the sample SD s but then we need to widen the intervals to account for the uncertainty in s 500 BAD confidence intervals for u 5 unknown 500 confidence intervals for u 6 unknown I I I I I L t I j I I r I 1 I r r I I l L I I r I I I J I I I I I 1 I 1 I I I T I El I I I I E E a i I I I I L J I I E E a E I I I l l E 1 I I 1 E E i E E E a E E E E E E E l L f E E E E E E 1 E E E E E E a E E a E i E i E E E E F E E l E a 1 1 E E E E E E a E E E Z 5 E E E E 4 E 5 a j E f E E 39l I I I I I I I I i iiiiii iiiiii iiiiii iiiiii The Student t distribution If X17X27 Xn are iid normalmeanu SD0 X u i S tdfin 1 Discovered by William Gossett quotStudentquot who worked for Guiness In R use the functions pt qt and dt eg qt09759 returns 226 of 196 pt1969 pt 1969 returns 7 0918 of 095 4 2 0 2 4 The t interval lf X1 Xn are iid normalmeanu SD0 Xi tor2 n 1 sn is a1 or confidence interval for u ta2 n 1 is the 1 a2 quantile of the t distribution with n 1 degrees of freedom oc2 l l l l I l 2 4 toc2 n 1 In R qtO9759 for the case n10 Or5 Example 1 Suppose we have measured the og1o cytokine response of 10 mice and obtained the following numbers Data 02 13 14 23 42 X 368 n 10 47 47 51 59 70 s 224 qt09759 226 95 confidence interval for u the population mean 368 i 226 x 224 v10 m 368 i 160 21 53 Example 2 Suppose we have measured by RealTimePCR the logo expression of a gene in 3 tissue samples and obtained the following numbers Data 117 635 776 x 509 n 3 s 347 qtO9752 430 95 confidence interval for u the population mean 509 i 430 x 347 3 m 509 i 862 35 137 I iQSACI Example 3 Suppose we have weighed the mass of tumor in 20 mice and obtained the following numbers Data 349 285 343 384 296 X 307 n 20 282 253 321 S 606 qto97519 209 95 confidence interval for u the population mean 307 i 209 x 606 V20 m 307 i 284 279 335 One last point I hate the phrase We are 95 confident that I also dislike the word statistically
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