1STAT 10, UCLA, Ivo DinovSlide 1UCLA STAT 10Introduction toStatistical ReasoningInstructor: Ivo Dinov, Asst. Prof. In Statistics and NeurologyTeaching Assistants: Yan Xiong, Will Anderson,UCLA StatisticsUniversity of California, Los Angeles, Winter 2002http://www.stat.ucla.edu/~dinov/STAT 10, UCLA, Ivo DinovSlide 2Chapter 21: Confidence IntervalsIntroductionMeansProportionsComparing 2 meansComparing 2 proportionsHow big should my study be?STAT 10, 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 10, 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, 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 z standard errors±STAT 10, UCLA, Ivo DinovSlide 5 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 process of taking samples effects the estimates and the CI’s. 1000 samples of size 10 obs’s from a Normal(m=24.83, s=0.005) distributions with their 95% CI’s.x24.833rd2nd1st100%100%100%SampleCoverageto dateTrue mean24.83o1000th999th95.2%95.2%Truemean24.8424.82)(262.2)(9xSExxSEtx±±±±====±±±±True mean almost alwayscaptured in the CI.STAT 10, 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 mean24.8424.82Most ofthe tableHow many of the previous samples contained the true mean?2STAT 10, UCLA, Ivo DinovSlide 7The Z and the T scores/values Remember Z = (X – mean)/SD ~ std. Normal (0, 1) We know how to read the Standard Normal (Z ) tables How about if the SD is unknown? We estimate it from the data, using the sample-SD Then we’d like to do the same we did for normal standardization –T = (X – mean) / sample_SD But is T standard normally distributed? Almost! T = (X–mean)/sample_SD ~ Student’s T (df=n-1)STAT 10, UCLA, Ivo DinovSlide 8The Z and the T scores/values T = (X–mean)/sample_SD ~ Student’s T (df=n-1)024- 2- 4df = ×[i.e., Normal(0,1)]df = 5df = 2STAT 10, UCLA, Ivo DinovSlide 9Confidence 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 10, UCLA, Ivo DinovSlide 1080% 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)Effect of increasing the confidence levelConfidenceLevelIncreaseIncreases the sizeof the CIWhy?STAT 10, UCLA, Ivo DinovSlide 11Effect of increasing the sample sizeTo double the precision we need four times as many observations. Passage 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.IncreaseSampleSizeDecreases the sizeof the CISTAT 10, 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 10, 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 proportionSTAT 10, UCLA, Ivo DinovSlide 14Example – higher blood thiol concentrationswith rheumatoid arthritisTABLE 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 0.44049Research 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 10, UCLA, Ivo DinovSlide 15Thiol concentration (mmol)1.5 2.0 2.5 3.0 3.5 4.0 4.5NormalRheumatoidFigure 8.4.1Dot 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 disease?STAT 10, UCLA, Ivo DinovSlide 16CI’s for difference between meansConfidence Interval for a difference between population means ( ):Difference between sample means t standard errors of the differenceor±x 1− x 2±t se(x 1− x 2)21µ−µSTAT 10, UCLA, Ivo DinovSlide 17CI’s for difference between proportionsConfidence Interval for a difference between population proportions ( ):Difference between sample proportions z standard errors of the differenceor±ˆ p 1−ˆ p 2±z se(ˆ p 1−ˆ p 2)21pp −But how do we compute the SE( ) for different cases?2ˆ1ˆpp−−−−STAT 10, UCLA, Ivo DinovSlide 18Sample size - proportion For a 95% CI,  Sample size for a desired margin of error:For a margin of error no greater than m, use a sample size of approximately p* is a guess at the value of the proportion -- err on the side of being too close to 0.5 z is the multiplier appropriate for the confidence level m is expressed as a proportion (between 0 and 1), not a