CHAPTER 8: SAMPLING DISTRIBUTIONS By: Wandi Ding 1 8.1: DISTRIBUTION OF THE SAMPLE MEAN Sampling distribution: the sampling distribution of a statistic is a probability distribution for all possible values of the statistic computed from a sample of size n. Sampling distribution of the sample mean: The sampling distribution of the sample mean x is the probability distribution of all possible values of the random variable computed from a sample of size n from a population with mean and standard deviation . The idea is as follows obtaining the sampling distribution: Step 1: obtain a simple random sample of size n Step 2: compute the sample mean . Step 3: assuming that we are sampling from a finite population, repeat steps 1 and 2 until all distinct simple random samples of n have been obtained. Note: once a particular sample is obtained, it cannot be obtained a 2 second time. 8.1: DISTRIBUTION OF THE SAMPLE MEAN Ex: the below is to only illustrate the sampling distribution : a small population including 4 individuals is shown below and also list all sample with size=3. Population: 10 12 18 20 and population mean 15 and 4 . 12 all samples with size=3 listed below: Sample Sample mean x Sample 1 10 12 18 13.3 Sample 2 10 12 20 14 Sample 3 12 18 20 16.7 Sample 4 10 18 20 16 3 8.1: DISTRIBUTION OF THE SAMPLE MEAN The mean and standard deviation of the sampling distribution of x : Suppose that a simple random sample of size n is drawn from a large population with mean and standard deviation . The sampling distribution of x will have mean and standard deviation x called x n the standard error of the mean. (there is a note on the bottom of page 381, keep in mind! And then using this correction to redo our example again.) Ex: for the previous example, we calculate the standard deviation of the sampling mean is: 4.12 x 2.38 n 3 also called standard 4 error of the mean. 8.1: DISTRIBUTION OF THE SAMPLE MEAN Comment: If a random variable X is normally distributed, the distribution of the sample mean, x , is normally distributed. Its mean is the same as the population mean and the standard deviation of sample mean is x n Central limit theorem: Regardless of the shape of the underlying population, the sampling distribution of sample mean, x , becomes approximately normal as the sample size, n, increases. Comments: 1. when parent distribution is normally distributed, the distribution of the sample means will be normally distributed for any sample size, n. 5 8.1: DISTRIBUTION OF THE SAMPLE MEAN 2. if the parent distribution is not normal, the sample size n≥30, the distribution of the sample means is kind of normal, as n increases, the distribution of sample n will approach normal. Let’s take a look an question and some figures on page 386. 6 .