1Stat 13, UCLA, Ivo DinovSlide 1UCLA STAT 13Introduction toStatistical Methods for the Life and Health SciencesInstructor: Ivo Dinov, Asst. Prof. of Statistics and NeurologyTeaching Assistants:Fred Phoa, Kirsten Johnson, Ming Zheng & Matilda HsiehUniversity of California, Los Angeles, Fall 2005http://www.stat.ucla.edu/~dinov/courses_students.htmlStat 13, UCLA, Ivo DinovSlide 2Sampling Distribution for the Mean and Introduction to Confidence IntervalsStat 13, UCLA, Ivo DinovSlide 3Quantitative Dataz More complex than dichotomous dataz Sample and populations for quantitative data can be described in various ways: mean, median, standard deviation (each has it's own sampling distribution.)Stat 13, UCLA, Ivo DinovSlide 4Sampling Distribution of yz Recall: is used to estimate µz Question: How close to µis ? Before we can answer this we need to define the probability distribution that describes sampling variability ofyyyPopulationSample(n)Sample(n)Sample(n)sy,sy,sy,σµ,Stat 13, UCLA, Ivo DinovSlide 5Sampling Distribution ofyz Two really important facts: The average of the sampling distribution of is µ Notation: The standard deviation of the sampling distribution of is Notation: Note: As n Æ , gets smaller Why? Look at the formula Intuitively does this make sense?yµµ=yynσnyσσ=∞yσStat 13, UCLA, Ivo DinovSlide 6Sampling Distribution ofyz Theorem 5:1 p.159 (mean of the sampling distribution of = the population mean) (standard deviation (sd) of the sampling distribution of = the population SD divided by ) Shape: If the distribution of Y is normal the sampling distribution ofis normal. Central Limit Theorem (CLT) - If n is large, then the sampling distribution of is approximately normal, even if the population distribution of Y is not normal.yµµµ=ynnσnyσσ=yyy2Stat 13, UCLA, Ivo DinovSlide 7Central Limit Theorem (CLT)z No matter what the distribution of Y is, if n is large enough the sampling distribution of will be approximately normally distributed HOW LARGE??? Rule of thumb n > 30.zThe closeness of to µdepends on the sample sizez The more skewed the distribution, the larger n must be before the normal distribution is an adequate approximation of the shape of sampling distribution ofz Why? yyyStat 13, UCLA, Ivo DinovSlide 8Example: Appletshttp://socr.stat.ucla.edu/Applets.dir/SamplingDistributionApplet.htmlCentral Limit Theorem (CLT)Stat 13, UCLA, Ivo DinovSlide 9Application to DataExample: LA freeway commuters (mean/SD commute time:µ= 130σ= 20Suppose we randomly sample 4 drivers.FindFindyµyσ130==µµy10420===nyσσStat 13, UCLA, Ivo DinovSlide 10130Y150110130y140120z Visually:Application to DataStat 13, UCLA, Ivo DinovSlide 11130Y150110Example: LA freeway commuters (cont’)Suppose we randomly select 100 driversy132128130As n gets larger the variability in the sampling distribution gets smaller.Application to DataStat 13, UCLA, Ivo DinovSlide 12Example: LA freeway commuters (cont’)Suppose we want to find the probability that the mean of the 100 randomly selected drivers is more than 135 mmHgz First step: Rewrite with notation!~ N(130,2)z Second step: Identify what we are trying to solve!y)135( >yPApplication to Data3Stat 13, UCLA, Ivo DinovSlide 13z Third step: Standardizez Fourth Step: Use the standard normal table to solve1 – 0.9938 = 0.0062If we were to choose many random samples of size 100 from the population about 0.6% would have a mean SBP more than 135 mmHg.)5.2(2130135)135( >=⎟⎟⎠⎞⎜⎜⎝⎛−>−=> ZPyPyPyyσµApplication to DataStat 13, UCLA, Ivo DinovSlide 14Example: LA freeway commuters (cont’)n ()135125 << YP yσ 4 ()3830.05.05.0 =<<− ZP 10420= 10 ()5704.079.079.0 =<<− ZP 32.61020= 20 ()7372.012.112.1 =<<− ZP 47.42020= 50 ()9232.077.177.1 =<<− ZP 83.25020= The mean of a larger sample is not necessarily closer to µ, than the mean of a smaller sample, but it has a greater probability of being closer to µ.Therefore, a larger sample provides more information about the population meanApplication to DataStat 13, UCLA, Ivo DinovSlide 15Notationz Notation: mean standard deviation Population µ σ Sample y s Sampling Distribution of y yµ yσ=nσ Stat 13, UCLA, Ivo DinovSlide 16Other Aspects of Sampling Variability• Sampling variability in the shape• Sampling variability in the sample standard deviationOverall: If n is large, s , the shape of each sample will be close to the shape of the population, and the shape of the sampling distribution of will approach a normal distribution.σySample(n)sy,µ, σPopulationSample(n)sy,Sample(n)sy,Stat 13, UCLA, Ivo DinovSlide 17Statistical Estimationz This will be our first look at statistical inferencez Statistical estimation is a form of statistical inference in which we use the data to: determine an estimate of some feature of the population assess the precision of the estimateStat 13, UCLA, Ivo DinovSlide 18Example: A random sample of 45 residents in LA was selected and IQ was determined for each one. Suppose the sample average was 110 and the sample standard deviation was 10.What do we know from this information?= 110S = 10yStatistical Estimation4Stat 13, UCLA, Ivo DinovSlide 19z The population IQ of LA residents could be described by µand σ110 is an estimate of µ10 is an estimate of σz We know there will be some sampling error affecting our estimates Not necessarily in the measurement of IQ, but because only 45 residents were sampledStatistical EstimationStat 13, UCLA, Ivo DinovSlide 20z QUESTION: How good is as an estimate of µ? z To answer this we need to assess the reliability of our estimatez We will focus on the behavior of in repeated sampling Our good friend, the sampling distribution ofyyyStatistical EstimationStat 13, UCLA, Ivo DinovSlide 21The Standard Error of the Meanz We know the discrepancy between and from sampling error can be described by the sampling distribution of , which uses to measure the variability Recall:z Is there a problem with obtaining from our data?z What seems like a good estimate for ?is an estimate forCalled the standard error of the meanµyyyσnyσσ=yσyσnsnσStat 13, UCLA, Ivo DinovSlide 22The Standard Error of the Meanz Notation for the standard error of the mean Sometimes referred to as the standard error (SE)z Round to two significant digitsnsSEy=Stat 13, UCLA, Ivo DinovSlide
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