1STAT 13, UCLA, Ivo DinovSlide 1UCLA STAT 13Introduction toStatistical Methods for the Life and Health Sciences!Instructor: Ivo Dinov, Asst. Prof. In Statistics and Neurology!Teaching Assistants: Sovia Lau, Jason ChengUCLA StatisticsUniversity of California, Los Angeles, Fall 2003http://www.stat.ucla.edu/~dinov/courses_students.htmlSTAT 13, UCLA, Ivo DinovSlide 2Chapter 8: Confidence Intervals!Introduction!Means!Proportions!Comparing 2 means!Comparing 2 proportions!How big should my study be?STAT 13, UCLA, Ivo DinovSlide 3(95% confiden24.820 24.825 24.830 24.835x ± 2 se’sx ± 2.36 se’s24.820 24.825 24.830 24.835(95% confidenx ± 2.09 se’sx ± 2 se’s20 replicated measurements to estimate the speed of light. Obtained by Simon Newcomb in 1882, by using distant (3.721 km) rotating mirrors.Passage time (10-6seconds)Using only 8Of the 20observationsUsing all 20observationsEstimates of the speed of light95% ConfidenceInterval shrinks?!?95%Confidence IntervalSTAT 13, UCLA, Ivo DinovSlide 4A 95% confidence interval! A type of interval that contains the true value of a parameter for 95% of samples taken is called a 95%confidence interval for that parameter, the ends of the CI are called confidence limits.!(For the situations we deal with) a confidence interval (CI) for the true value of a parameter is given byestimate t standard errorsTABLE 8.1.1 Value of the Multiplier, t, for a 95% CIdf : 7 8 9 10 11 12 13 14 15 16 17t : 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110df :18192025303540455060 t : 2.101 2.093 2.086 2.060 2.042 2.030 2.021 2.014 2.009 2.000 1.960∞±STAT 13, UCLA, Ivo DinovSlide 524.83o1000th999th95.2%95.2%True mean24.8424.82Figure 8.1.2 Samples of size 10 from a Normal(µ=24.83, s=.005) distribution and their 95% confidence intervals for µ..24.833rd2nd1st100%100%100%SampleCoverageto dateTrue mean! CI are constructed using the sample and s=SE. But diff. samples yield diff. estimates and " diff. CI’s?!?! Below is a computer simulationshowing how the process of taking samples effects the estimates and the CI’s.x)(262.2)(9xSExxSEtx ±=±True mean almost always captured in the CI.STAT 13, UCLA, Ivo DinovSlide 624.83o24.83500th100th10th9th8th7th6th5th4th3rd2nd1st1000th999th998th997th996th995th994th993rd992nd991st502nd501st96.0%94.0%90.0%88.9%100%100%100%100%100%100%100%100%95.2%95.2%95.2%95.2%95.2%95.2%95.2%95.2%95.2%95.2%96.0%96.0%............................................................SampleCoverageto dateTrue meanTrue mean24.8424.82Figure 8.1.2 Samples of size 10 from a Normal(µ=24.83, s=.005) distribution and their 95% confidence intervals for µ..From Chance En counters by C.J. Wi ld and G.A.F. Se ber, © Joh n Wiley & Sons, 1999.Most ofthe tableHow many of the previous samples contained the true mean?2STAT 13, UCLA, Ivo DinovSlide 7Confidence Interval for the true (population) mean µ:sample mean t standard errorsor±x ± t se(x ), where se(x ) =sxn and df = n −1Summary - CI for population meanTABLE 8.1.1 Value of the Multiplier, t, for a 95% CIdf : 7 8 9 10 11 12 13 14 15 16 17t : 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 2.120 2.110df :18192025303540455060 t : 2.101 2.093 2.086 2.060 2.042 2.030 2.021 2.014 2.009 2.000 1.960∞STAT 13, UCLA, Ivo DinovSlide 8STAT 13, UCLA, Ivo DinovSlide 9Output from other statistics packagesT Confidence IntervalsVariable N Mean StDev SE Mean 95.0 % CIPassage Times 20 24.8286 0.0051 0.0011 ( 24.8262, 24.8309)Passage TimesMean 24.82855Standard Error 0.0011459Standard Deviation 0.0051245Count 20Confidence Level(95.0%) 0.0023983. . . . . . . .. . . . . . . .Minitab OutputSelected Excel OutputMinitab from menus:Stat Basic Statistics 1-Sample tCheck “Confidence interval” in dialogue boxExcel from menus:Tools Data AnalysisChoose “Descriptive Statistics”,Check “Summary” and “ConfidenceLevel for Mean” in dialogue boxxCI =24.82855 ± 0.0023983 ± term for CIFigure 8.2.1 Computer output for Newcomb's passage-time data.STAT 13, UCLA, Ivo DinovSlide 10Effect of increasing the confidence level80% CI, x ± 1.282 se(x)90% CI, x ± 1.645 se(x)95% CI, x ± 1.960 se(x)99% CI, x ± 2.576 se(x)Figure 8.1.3The greater the confidence level, the wider the intervalFrom Chance Encounters by C.J. Wild and G.A.F. Seber, © John Wiley & Sons, 2000.80% CI, x ± 1.282 se(x)90% CI, x ± 1.645 se(x)99% CI, x ± 2.576 se(x)95% CI, x ± 1.960 se(x)ConfidenceLevelIncreaseIncreases the sizeof the CIWhy?STAT 13, UCLA, Ivo DinovSlide 11Effect of increasing the sample sizePassage timen = 90 data pointsn = 40 data pointsn = 10 data points24.83 24.8424.82Figure 8.1.4 Three random samples from a Normal(µ=24.83, s =.005) distribution and their 95% confidence intervals for µ.From Chance Encounters by C.J. Wild and G.A.F. Seber, © John Wiley & Sons, 2000,To double the precision we need four times as many observations. IncreaseSampleSizeDecreases the sizeof the CISTAT 13, UCLA, Ivo DinovSlide 12Why in sample-size CI?Confidence Interval for the true (population) mean µ:sample mean t standard errorsor±x ± t se(x ), where se(x ) =sxn and df = n −1↑↓3STAT 13, UCLA, Ivo DinovSlide 13Confidence Interval for the true (population) proportion p:sample proportion z standard errors±or ˆ p ± z se(ˆ p ), where se(ˆ p ) =ˆ p (1 −ˆ p )nCI for a population proportion, Section 7.3.STAT 13, UCLA, Ivo DinovSlide 14TABLE 8.4.1 Thiol Concentration (mmol)Normal Rheumatoid 1.84 2.811.92 4.061.94 3.621.92 3.271.85 3.271.91 3.762.07Sample size 7 6Sample mean 1.92143 3.46500Sample standard deviation 0.07559 0.44049Example – higher blood thiol concentrationsassociated with rheumatoid arthritis?!?Research question:Is the change in the Thiol statusin the lysate of packed blood cells substantial to be indicativeof a non trivial relationship between Thiol-levels and rheumatoid arthritis?STAT 13, UCLA, Ivo DinovSlide 15Thiol concentration (mmol)1.5 2.0 2.5 3.0 3.5 4.0 4.5NormalRheumatoidFigure 8.4.1 Dot plot of Thiol concentration data.Example – higher blood thiol concentrationswith rheumatoid arthritisTwo groups of subjects are studied: 1. NC (normal controls)2. RA (rheumatoid arthritis).Observations: 1. The avg. levels of thiol seem diff. in NC & RA2. NC and RA groups are separated completely.Question: Is there statistical evidence that thiol-level correlates withthe
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