SAMPLING DISTRIBUTION OF MEANS & PROPORTIONS SAMPLING AND SAMPLING VARIATION Sample Knowledge of students No. of red blood cells in a person Length of the life of electric bulbs Population Population census whole population Repeat the same study, under exactly similar conditions, we will not necessarily get identical results. Example: In a clinical trail of 200 patients we find that the efficacy of a particular drug is 75% If we repeat the study using the same drug in another group of similar 200 patients we will not get the same efficacy of 75%. It could be 78% or 71%. If we repeat the study using the same drug in another group of similar 200 patients we will not get the same efficacy of 75%. It could be 78% or 71%. Different results from different trails though all of them conducted under the same conditions Different results from different trails though all of them conducted under the same conditions Example: If two drugs have the same efficacy then the difference between the cure rates of these two drugs should be zero. But in practice we may not get a difference of zero. If we find the difference is small say 2%, 3%, or 5%, we may accept the hypothesis that the two drugs are equally effective. On the other hand, if we find the difference to be large say 25%, we would infer that the difference is very large and conclude that the drugs are not of equally efficacy. Example: If we testing the claim of pharmaceutical company that the efficacy of a particular drug is 80%. We may accept the companys claim if we observe the efficacy in the trail to be 78%, 81%, 83% or 77%. We may accept the companys claim if we observe the efficacy in the trail to be 78%, 81%, 83% or 77%. But if the efficacy in trail happens to be 50%, we would have good cause to feel that true efficacy cannot be 80%. But if the efficacy in trail happens to be 50%, we would have good cause to feel that true efficacy cannot be 80%. And the chance of such happening must be very low. We then tend to dismiss the claim that the efficacy of the drug is 80%. And the chance of such happening must be very low. We then tend to dismiss the claim that the efficacy of the drug is 80%. THEREFORE WHILE TAKING DECISIONS BASED ON EXPERIMENTAL DATA WE MUST GIVE SOME ALLOWANCE FOR SAMPLING VARIATION . WHILE TAKING DECISIONS BASED ON EXPERIMENTAL DATA WE MUST GIVE SOME ALLOWANCE FOR SAMPLING VARIATION . VARIATION BETWEEN ONE SAMPLE AND ANOTHER SAMPLE IS KNOWN AS SAMPLING VARIATION. VARIATION BETWEEN ONE SAMPLE AND ANOTHER SAMPLE IS KNOWN AS SAMPLING VARIATION. Inference extension of results obtained from an experiment (sample) to the general population use of sample data to draw conclusions about entire population Parameter number that describes a population Value is not usually known We are unable to examine population Statistic number computed from sample data Estimate unknown parameters Computed to estimate unknown parameters Mean, standard deviation, variability, etc.. Notations population mean sample mean SAMPLING DISTRIBUTION The sample distribution is the distribution of all possible sample means that could be drawn from the population. SAMPLING DISTRIBUTIONS What would happen if we took many samples of 10 subjects from the population? Steps: 1.Take a large number of samples of size 10 from the population 2.Calculate the sample mean for each sample 3.Make a histogram of the mean values 4.Examine the distribution displayed in the histogram for shape, center, and spread, as well as outliers and other deviations How can experimental results be trusted? If is rarely exactly right and varies from sample to sample, how it will be a reasonable estimate of the population mean ? How can we describe the behavior of the statistics from different samples? E.g. the mean value Very rarely do sample values coincide with the population value (parameter). The discrepancy between the sample value and the parameter is known as sampling error, when this discrepancy is the result of random sampling. Fortunately, these errors behave systematically and have a characteristic distribution. A sample of 3 students from a class a population of 6 students and measure students GPA StudentGPA Susan2.1 Karen2.6 Bill2.3 Calvin1.2 Rose3.0 David2.4 Draw each possible sample from this population: Susan 2.1 Karen 2.6 Bill 2.3 Rose 3.0 David 2.4 Calvin 1.2 With samples of n = 3 from this population of N = 6 there are 20 different sample possibilities: Note that every different sample would produce a different mean and s.d., ONE SAMPLE = Susan + Karen +Bill / 3 = / 3 = 7.0 / 3 = 2.3 Standard Deviation: ( ) 2 =.2 2 =.04 ( ) 2 =.3 2 =.09 ( ) 2 = 0 2 = 0 s 2 =.13/3 and s = =.21 So this one sample of 3 has a mean of 2.3 and a sd of.21 What about other samples? A SECOND SAMPLE = Susan + Karen + Calvin = = 1.97 = 1.97 SD =.58 20 th SAMPLE = Karen + Rose + David = = 2.67 SD =.25 Assume the true mean of the population is known, in this simple case of 6 people and can be calculated as 13.6/6 = =2.27 The mean of the sampling distribution (i.e., the mean of all 20 samples) is 2.30. Sample mean is a random variable. If the sample was randomly drawn, then any differences between the obtained sample mean and the true population mean is due to sampling error. Any difference between and is due to the fact that different people show up in different samples If is not equal to , the difference is due to sampling error. Sampling error is normal, it is to-be-expected variability of samples What is a Sampling Distribution? A distribution made up of every conceivable sample drawn from a population. A sampling distribution is almost always a hypothetical distribution because typically you do not have and cannot calculate every conceivable sample mean. The mean of the sampling distribution is an unbiased estimator of the population mean with a computable standard deviation. LAW OF LARGE NUMBERS If we keep taking larger and larger samples, the statistic is guaranteed to get closer and closer to the parameter value. If we keep taking larger and larger samples, the statistic is guaranteed to get closer and closer to the parameter value. N = 1N = 2 N = 10 N = 25 Properties of sampling distributions Central Limit Theorem If all possible random samples, each the size of your sample, were taken from any population then the sampling distribution of sample means will have: a mean equal to the population mean a standard deviation equal to The sampling distribution will be normally distributed IF EITHER: the parent population from which you are sampling is normally distributed OR IF the sample size is greater than n=30. ILLUSTRATION OF SAMPLING DISTRIBUTIONS Draw 500 different SRSs. What happens to the shape of the sampling distribution as the size of the sample increases? 500 Samples of n = 2 500 Samples of n = 4 500 Samples of n = 6 500 Samples of n = 10 500 Samples of n = 20 Key Observations As the sample size increases the mean of the sampling distribution comes to more closely approximate the true population mean, here known to be = 3.5 AND-this critical-the standard error-that is the standard deviation of the sampling distribution gets systematically narrower. Three main points about sampling distributions Probabilistically, as the sample size gets bigger the sampling distribution better approximates a normal distribution. The mean of the sampling distribution will more closely estimate the population parameter as the sample size increases. The standard error (SE) gets narrower and narrower as the sample size increases. Thus, we will be able to make more precise estimates of the whereabouts of the unknown population mean. ESTIMATING THE POPULATION MEAN We are unlikely to ever see a sampling distribution because it is often impossible to draw every conceivable sample from a population and we never know the actual mean of the sampling distribution or the actual standard deviation of the sampling distribution. But, here is the good news: We can estimate the whereabouts of the population mean from the sample mean and use the samples standard deviation to calculate the standard error. The formula for computing the standard error changes, depending on the statistic you are using, but essentially you divide the samples standard deviation by the square root of the sample size. THE MEAN OF THE SAMPLING DISTRIBUTION The mean of a sampling distribution ( ) is made up of all possible SRSs of the same size as your sample. Its mean will equal the population mean from which it was drawn. The distribution of sample means will be normally distributed, centered at the population mean with a standard deviation of the sampling distribution, called the standard error (SE). Dont get confuse with the terms of STANDARD DEVEIATION and and STANDARD ERROR Quantifying Uncertainty Standard deviation: measures the variation of a variable in the sample. Technically, Standard error of mean is calculated by: Standard deviation versus standard error The standard deviation (s) describes variability between individuals in a sample. The standard error describes variation of a sample statistic. . The standard deviation describes how individuals differ. The standard error of the mean describes the precision with which we can make inference about the true mean. Standard error of the mean Standard error of the mean (sem): Comments: n = sample size even for large s, if n is large, we can get good precision for sem always smaller than standard deviation (s) Proportions A proportion or percentage is a mean: it is a mean of a variable that takes on the values 0 and 1. The event of interest is coded 1. The CLT then applies to proportions as it does to means. For a 0/1 variable, the population is necessarily not normally distributed, but the CLT says that for a proportion calculated from a large sample the sampling distribution will be normally distributed. Notation p = population proportion = sample proportion n = sample size CLT suggests: mean of sampling distribution of proportion p standard deviation of sampling distribution of proportion For a 0/1 variable, the standard deviation simplifies to a simple function of the proportion ones in the population: The standard deviation of the sampling distribution then simplifies as follows: Normality of Sampling Distributions In small samples, the sampling distribution of a proportion will not be normally shaped because the population of a normal. Rule of thumb: the sampling distribution is close enough to normal to use the normal table if np 10 and n(1-p) 10 Otherwise, we cannot do the problem with the normal table. SLOGAN TO REMEMBER Sample Mean + Sampling Error = The Population Mean Some Sample Characteristic + Sampling Error = The Population Characteristic Two Steps in Statistical Inferencing Process 1.Calculation of confidence intervals from the sample mean and sample standard deviation within which we can place the unknown population mean with some degree of probabilistic confidence 2.Compute test of statistical significance (Risk Statements) which is designed to assess the probabilistic chance that the true but unknown population mean lies within the confidence interval that you just computed from the sample mean. So, first we calculate confidence limits and then test for statistical significance, which is the proba- bility of mu being within the CIs we computed. Both these steps are required when making inferences about the whereabouts of the unknown population mean. Both the calculation of confidence intervals and then the calculation of a measure of statistical likelihood -- are based on the probabilistic patterns of a sampling distribution. Together, the confidence limits and statistical test tells us the probability as to what would happen IF we sampled the population not once but an infinite number of times. That is, we are sampling from a sampling distribution.This kind of inferencing is the hallmark of statistics. What we want to do now is to take the next step, to learn how to substantiate our conclusions -- to learn how to back up our conclusions with analyses that will reflect how much confidence we should have that our estimate of say the mean of the population -- which is being estimated from our sample -- is at or close to the true population mean.