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:Brandi Shanata & Tiffany HeadUniversity of California, Los Angeles, Fall 2007http://www.stat.ucla.edu/~dinov/courses_students.htmlStat 13, UCLA, Ivo DinovSlide 2Chapter 5Sampling DistributionsStat 13, UCLA, Ivo DinovSlide 3Sampling Distributionsz Definition: Sampling Variability is the variability among random samples from the same population.z A probability distribution that characterizes some aspect of sampling variability is called a sampling distribution. tells us how close the resemblance between the sample and the population is likely to be.z We typically construct a sampling distribution for a statistic. Every statistics has a sampling distribution.Stat 13, UCLA, Ivo DinovSlide 4The Meta-Experimentz All the possible samples that might be drawn from the population (infinity repetitions). In other words if we were to repeatedly take samples of the same size from the same population, over and over.PopulationSample(n)Sample(n)Sample(n)Stat 13, UCLA, Ivo DinovSlide 5The Meta-Experimentz Meta-experiments are important because probability can be interpreted as the long run relative frequency of the occurrence of an event.z Meta-experiments also let us visualize sampling distributions. and therefore understand the variability among the many random samples of a meta-experiment.Stat 13, UCLA, Ivo DinovSlide 6Dichotomous Observationsz Dichotomous - two outcomes (yes or no, good or evil, etc…) z We use the following notation for a dichotomous outcomeP population proportionsample proportionz The big question is how close is to P?z To determine this we need to examine the sampling distribution ofz What we want to know is: if we took many samples of size n and observed each time, how would those values of be distributed around p?pˆpˆpˆpˆ2Stat 13, UCLA, Ivo DinovSlide 7Dichotomous ObservationsExample: Suppose we would like to estimate the trueproportion of male students at UCLA. We could take a random sample of 50 students and calculate the sample proportion of males.z What is the correct notation for: the true proportion of males? the sample proportion of males?z Suppose we repeat the experiment over and over. Would we get the same proportion of males for the second sample?Stat 13, UCLA, Ivo DinovSlide 8Reece’s Pieces ExperimentExample: Suppose we would like to estimate the true proportion of orange reece’s pieces in a bag. To investigate we will take a random sample of 10 reece’s pieces and count the number of orange. Next we will make an approximation to a sampling distribution with our class results.What you need to calculate: the number of orange the sample proportion of orange (number of orange/10)Stat 13, UCLA, Ivo DinovSlide 9An Application of a Sampling DistributionExample: Mendel's pea experiment. Suppose a tall offspring is the event of interest and that the true proportion of tall peas (based on a 3:1 phenotypic ratio) is 3/4 or p = 0.75. If we were to randomly select samples with n = 10 and p = 0.75 we could create a probability distribution as follows:pˆ Number Tall Number Dwarf Probability0.0 0 10 0.000 0.1 1 9 0.000 0.2 2 8 0.000 0.3 3 7 0.003 0.4 4 6 0.016 0.5 5 5 0.058 0.6 6 4 0.146 0.7 7 3 0.250 0.8 8 2 0.282 0.9 9 1 0.188 1.0 10 0 0.056 Lab_Mendel_Pea_Experiment.html(work out in discussion/lab)Validate using:http://socr.stat.ucla.edu/Applets.dir/Normal_T_Chi2_F_Tables.htmE.g., B(n=10, p=0.75, a=6, b=6)=0.146Stat 13, UCLA, Ivo DinovSlide 10pˆ Number Tall Number Dwarf Probability0.0 0 10 0.000 0.1 1 9 0.000 0.2 2 8 0.000 0.3 3 7 0.003 0.4 4 6 0.016 0.5 5 5 0.058 0.6 6 4 0.146 0.7 7 3 0.250 0.8 8 2 0.282 0.9 9 1 0.188 1.0 10 0 0.056 z What is the probability that 5 are tall and 5 are dwarf?P(5 tall and 5 dwarf) = P( = 5/10)= P( = 0.5)= 0.058pˆpˆAn Application of a Sampling DistributionStat 13, UCLA, Ivo DinovSlide 11z If we think about this in terms of a meta-experiment and we sample 10 offspring over and over, about 5.8% of the 's will be 0.5. This is the sampling distribution of sample proportion of tall offspring is the distribution of in repeated samples of size 10.z If we take a random sample of size 10, what is the probability that six or more offspring are tall?P( > 0.6) = 0.146 + 0.250 + 0.282 + 0.188 + 0.056= 0.922 pˆpˆAn Application of a Sampling DistributionStat 13, UCLA, Ivo DinovSlide 12z This table could also be represented as a histogram with probability on the y-axis and proportion on the x-axis. easier to draw these by hand1.00.90.80.70. 60.50. 40. 30.20.10. 00.30.20.10.0proportionProbabilityAn Application of a Sampling Distribution3Stat 13, UCLA, Ivo DinovSlide 13Relationship to Statistical Inferencez We can also use our sampling distribution of to estimate how much sampling error there is within 5 percentage points of p. Because we knew p from the previous example (p=0.75), we might want to estimate:P(0.7 < < 0.8) = 0.250 + 0.282 = 0.532 There is a 53% chance that for a sample of size 10, will be within +0.05 of p.This seems a little crazy, why?pˆpˆ Number Tall Number Dwarf Probability0.0 0 10 0.000 0.1 1 9 0.000 0.2 2 8 0.000 0.3 3 7 0.003 0.4 4 6 0.016 0.5 5 5 0.058 0.6 6 4 0.146 0.7 7 3 0.250 0.8 8 2 0.282 0.9 9 1 0.188 1.0 10 0 0.056 pˆStat 13, UCLA, Ivo DinovSlide 14z So far we have been using p to determine the sampling distribution of . z Why sample for when we already know p? We don't need to know p to get a good estimate (this will come later). pˆpˆRelationship to Statistical InferenceStat 13, UCLA, Ivo DinovSlide 15Sample Sizez As n gets larger, will become a better estimate of p. z Just to show…*These calculations were done using the SOCR binomial distribution Calculator. http://www.socr.ucla.edu/Applets.dir/Normal_T_Chi2_F_Tables.htmE.g., B(n=20, p=0.75, a=0.7x20=14, b=0.8x20=16)=0.5606THE POINT: A larger sample improves the chance that is close to p. Caution: this doesn’t necessarily mean that the estimate will be closer to p, only that there is a better chance that it will be close to p.pˆN P(0.7 < pˆ < 0.8) 10 0.53 20 0.56 50 0.673 100 0.798 pˆStat 13, UCLA, Ivo DinovSlide 16Sampling Distribution for the Mean and Introduction to Confidence
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