Applied Business Statistics Week 5 Continued Sampling Distribution Today s Agenda Today Sampling Distribution Practice in Excel with Sampling Distribution Concepts Last 15 Mins Discuss Midterm Discuss Midterm Review Sampling Distribution A theoretical probabilistic distribution of a statistic i e mean for all possible draws of a specific sample size n A sampling distribution is a distribution of statistics obtained by selecting all the possible samples of a specific size from a population 3 To get a Sampling Distribution 1 Take a random sample of size n from this population 2 Compute the statistic e g the mean and record it Put the sample back 3 Take a 2nd 3rd random sample of size n calculating the mean each time 4 Plot the resulting sampling distribution a distribution of a statistic over repeated samples Sampling Distribution A sampling distribution is a distribution of a statistic over all possible samples Suppose we re interested in the resting heart rate of students in this classroom Here s how we would get a sampling distribution Sampling Distribution Population of 5 N 5 Sampling Distribution The distribution of statistics obtained by selecting all possible samples of a specific size from a population Statistic Value obtained from a sample 1 59 4 41 S4 S3 S1 cid 0 cid 0 3 S2 S5 Means Sample 1 1 2 3 4 2 5 Sample 2 1 2 3 5 2 75 Sample 3 1 2 4 5 3 0 Sample 4 1 3 4 5 3 25 Sample 5 2 3 4 5 3 5 Sampling Distribution Is the central concept of inferential statistics It is theoretical Meaning it is never obtained in reality by the researcher However because it contains all possible sample outcomes and is governed by the laws of probability it enables us to estimate the probability of any particular sample outcome 95 68 2 5 2 1 0 1 2 2 5 Sampling Error Standard Error SE Descriptive Statistics summary table from Excel SE of the sample mean Standard Error SE The higher the standard error the more uncertainty we have about the location of the true population mean SE of the sample mean Magnitude of standard error determined by 1 sample size and 2 the standard deviation of the population As n increases the standard error decreases Thus the higher the number of observations n the lower the SE The lower the SE the more confident you are that your sample mean is a good estimate of your population mean SE of the sample mean Standard Error SE Descriptive Statistics summary table from Excel SE formula or Variability of a sample selected from a population Each sample yields a different statistic which is what creates the distribution of sample means or sampling distribution cid 0 sampling error Each statistic will have some error difference between itself and the true population mean associated with it This is referred to as the sampling error Like any other distribution the sampling distribution also has a standard deviation Standard Error SE On your way out of class you ask 5 students how long in seconds it takes them to walk to the Student Union from JFF LL105 How confident can we be about the estimate of the true mean SE of the sample mean n 5 n 50 n 500 Sampling Error Poll Class of 175 students In a typical day about how much time to you spend watching television Sample mean 2 09 hours Sample standard deviation 1 644 hours Central Limit Theorem As n increases the distribution of sample means will become more normal in shape The central limit theorem tells us that sample averages are normally distributed if we have enough data This is true even if our original variables are not normally distributed The distribution of sample means approaches normal very rapidly By n 30 the distribution is almost perfectly normal 0 1 2 3 4 5 6 7 8 Sample Means 4 3 2 1 0 1 Central Limit Theorem 2 States that for a population with mean and standard deviation these three properties hold for the distribution of sample averages 3 The mean of the sampling distribution is identical to the population mean The standard deviation of the distribution of the sample averages is n or the standard error SE of the mean For n large enough in the limit as n the shape of the sampling distribution of means is approximately normal 0 1 2 3 4 5 6 7 8 Sample Means 4 3 2 1 0