Sampling theory chapter 3 sampling for proportions shalabh, iit kanpur page 4 ii srswr since the sample mean y is an unbiased estimator of the population mean y in case of srswr, so the sample proportion, ep ey y p, i. Since we often want to draw conclusions about something in a population based on only one sample, understanding how our sample statistics vary from sample to sample, as captured by the standard error, is. A study of stratified sampling in variance reduction techniques for parametric yield estimation mansour keramat, student member, ieee, and richard kielbasa. Now that weve got the sampling distribution of the sample mean down, lets turn our attention to finding the sampling distribution of the sample variance. Simple random sampling moulinath banerjee university of michigan september 11, 2012 1 simple random sampling the goal is to estimate the mean and the variance of a variable of interest in a nite population by collecting a random sample from it. We will usually denote probability functions asf and, in this case,fy which is strictly positive and a function of the random variabley, the number of successes observed in n trials. Construct the histogram of the sampling distribution of the sample variance. An estimator takes different values for different samples sampling variability. Construct the histogram of the sampling distribution of the sample variance construct the histogram of the sampling distribution of the sample median use the sampling distribution simulationjava applet at the rice virtual lab in statistics to do the following. Sampling distributions a sampling distribution acts as a frame of reference for statistical decision making. But we can also evaluate hypotheses by testing the sample variances.
The variance of estimates provides a measure of the degree of concentration of the sample estimates around the expected value. Normal one sample problem let be a random sample from where both and are unknown parameters. If x is the age of one person in the group and x is the average age of the two people in a sample, find the mean and variance of the sampling distribution of x. In general, when samples of size n are taken from a normal distribution with variance, the sampling distribution of the has a chisquare distribution with n1 degrees of freedom. Plot the distribution and record its mean and standard deviation. This follows for the same reasons as estimation for the discrete distribution.
Psunit iii lesson 2 finding the mean and variance of the. The variance of the empirical distribution is varnx en n x enx2 o en n x xn2 o 1 n xn i1 xi xn2 the only oddity is the use of the notation xn rather than for the mean. Figure 45 illustrates a case where the normal distribution closely approximates the binomial when p is small but the sample size is large. I then calculate the mean and standard deviation of the sample. Psy 320 cal state northridge 8 sampling distribution the distribution of a statistic over repeated sampling from a specified population. Let the median random variable x have values x and density gx. Mean and and standard deviation of a sampling distribution. We will return to a coin flipping survey where the outcomes are head. Sampling distribution of the sample mean video khan. This unit covers how sample proportions and sample means behave in repeated samples.
Using the expression of the variance of y and its estimate in case of srswr, the variance of p. The anova is used primarily when dealing with 3 or more groups of subjects, or 3 or more levels. The variance of the empirical distribution the variance of any distribution is the expected squared deviation from the mean of that same distribution. The variance of the sampling distribution of the sample mean. It measures the spread or variability of the sample estimate about its expected value in. The oneway anova in all of the tests so far, we determined whether or not some independent variable or treatment had an effect by performing tests on sample means. A sampling distribution shows every possible result a statistic can take in every possible sample from a population and how often each result happens. Turn in this output with the rest of the homework for unit 5.
I begin by discussing the sampling distribution of the sample variance when sampling from. Binomial probability function this function is of passing interest on our way to an understanding of likelihood and loglikehood functions. Sampling distribution of the sample variance chisquare distribution. The central limit theorem also tells us that the distribution of x can be approximated by the normal distribution if the sample size is large. Central limit theorem convergence of the sample means distribution to the normal distribution. We will now give an example of this, showing how the sampling distribution of x for the number of.
Sampling, measurement, distributions, and descriptive statistics sample means. Putting this information together with what we know about the mean and variance of the sample average we get 2 xn, n. Standard errors of mean, variance, and standard deviation. For the particular simulation performed the values in drawmeans, the observed variance is 0. Sampling distribution of difference between means d. A statistic computed from a random sample or in a randomized experiment is a random variable because the outcome depends on which individuals are included in the sample. Link to a calculator page from the ucla statistics department that will allow you to find the cdf or pdf for any chisquare distribution or any other discrete or. A discussion of the sampling distribution of the sample variance.
Mean and variance of sampling distributions of sample means the mean and variance of sampling distribution of sample means 10. That is, the variance of the sampling distribution of the mean is the population variance divided by n, the sample size the number of scores used to. It is a theoretical probability distribution of the possible values of some sample statistic that would occur if we were to draw all. The sampling distribution weconsiderprobabilitysamples, settingasideresponsebiasandnonresponse bias. Stratified sampling 3, 4, 6 is a variance reduction technique in. One test that does this is the analysis of variance anova. Mean and variance for any sample size n and a srs x1. Then, for any sample size n, it follows that the sampling distribution of x. Exercises the concept of a sampling distribution is perhaps the most basic concept in inferential statistics. Sampling, measurement, distributions, and descriptive statistics sampling distribution if we draw a number of samples from the same population, then compute sample statistics for statistics computed from a number of sample distributions.
The larger the size of a sample, the smaller the variance of the sample mean. Central limit theorem distribution mit opencourseware. In the last video, we learned about what is quite possibly the most profound idea in statistics, and thats the central limit theorem. Cochran, p15 the deviation of each sample estimate from the expected value is.
Introduce you to sampling weights methods for calculating variances and standard errors for complex sample designs general introduction to these topics weights are unique to research studies and data sets options for calculating variances and standard errors will vary by study overview 2 you will have a basic understanding of. Thus, the sampling distribution of x is of interest. Having understood what happens now with sampling variance, confidence intervals, all the rest of the design effects, lets go back and reconsider grouping, and add a couple of factors before extending our development of stratified random sampling. Sampling distribution of means and the central limit theorem 39 8. The sampling distribution of the mean refers to the pattern of sample. The sampling distribution of the sample mean duration. The sampling distribution of x we are able to show 2 ex and varx n. And the reason why its so neat is, we could start with any distribution that has a well defined mean and variance actually, i wrote the standard deviation here in the last video, that should be the mean, and lets say it has some variance. Sampling is a procedure, where in a fraction of the data is taken from a large. The probability distribution of the sample statistic is called the sampling distribution. A sampling distribution is abstract, it describes variability from sample to sample, not across a sample. X2xn from any population distribution with mean x and.
Psunit iii lesson 2 finding the mean and variance of the sampling distribution of means free download as powerpoint presentation. It is quite easy in this course, because it is beyond the scope. For example, modeling a bell shaped histogram of experimental frequencies by a gaussian pdf probability density function or a fishers pdf with four parameters. How do i calculate the pdf of these sample values, given that i know the population values. A study of stratified sampling in variance reduction. The sampling distribution of the sample variance youtube. Suppose i have drawn n samples from a population of known mean and variance for example, a normal distribution with mean zero and variance 1. In inferential statistics, it is common to use the statistic x to estimate.613 1017 655 1188 1380 1453 1470 546 925 647 909 255 1155 1319 840 811 1179 179 1033 650 1480 254 1232 802 127 846 379 870 536 1341 73 544 930 189 573 355 898 488 41 1017 1012 1354