Discussion Exercise 7 Spring 2010 Name Matthew Marck Section 0101 Understanding Sample Variability and the Central Limit Theorem Part 1 Sampling Distribution of Sample Means This week we are thinking about the sampling distribution of sample means NOT the distribution of observations The sampling distribution of sample means SDSM is really only a theoretical concept given that we rarely go out and actually sample a population again and again to create a distribution of sample means Let s go through the main concepts of the Sampling Distribution of Sample Means A First the spread to the distribution of sample means changes as the sample size changes Go to stat tamu edu west applets samplemean html This applet assumes that repeated samples are being taken from a population with a population mean of 100 Take 30 repeated samples for each sample size 1 4 9 16 and 25 click on the blue arrow to change the sample size click sample 30 times to get 30 samples Note samples of size 1 really aren t samples they are individual observations The applet will randomly draw the sample and calculate the mean of the samples and place a dot on the graph to show where that mean falls After collecting 30 samples write down the mean of the all the sample means and the standard deviation of the distribution of sample means Fill in the chart below Sample size samples Mean of samples Std dev of samples 1 30 99 04 1 937 4 30 100 0 9603 9 30 99 93 0 6449 16 30 100 1 0 5096 25 30 100 1 0 4279 The more common name for the standard deviation of sampling means is the Standard Error If the sampling distribution of sample means is normally distributed the standard error should be equal to the population standard deviation divided by the square root of n 1 On the graph below show how the spread to the standard deviation of the means decreases as n increases copy information from your simulation 2 What value would you predict for the standard deviation of samples when n 25 0 4379 sd sqrt 25 sd 2 1895 2 Does this simulation show that as your sample size increase the spread of the sampling distribution of sample means decreases as expected YES or NO B The KEY to this exercise is to recognize that if the population is normally distributed the sampling distribution of sample means will have the same approximate mean regardless of sample size BUT the spread to this SDSMs will decrease the standard error decreases as the sample size increases Now let s take a look at this same idea another way Go to http onlinestatbook com stat sim sampling dist index html Click on Begin The top graph is the population the 2nd graph is the sample taken from the population the 3rd graph is the sampling distribution of sample means SDSM when n 5 Click on Fit Normal Let s make the 4th graph the SDSM when n 25 change None to Mean and n 25 and click on Fit Normal Now just so you know what is going on click on Animated and watch the program simulate taking a sample of size 5 and a sample of size 25 Now let s speed it up Click on 1 000 Answer the following questions about the SDSM 3 Which of the following is NOT a characteristic of the sampling distribution of sample means 2 pts each a b c d the mean is zero and the standard deviation is one the distribution of values is obtained by means of repeated sampling the samples are all of size n the samples are all drawn from the same population 4 Assume that you have repeatedly taken samples of size 5 from a population of size 30 What can be said about the individual sample means a Each sample mean will equal the population mean b the sample means will vary but be close to the population mean c the mean of the means will equal zero d the sample mean will equal 5 5 T or F The standard error is a measure of sampling variability between repeated sample means Part 2 The Central Limit Theorem states that the sampling distribution of sample means will approach a normal distribution as size of your samples increase They key point here if your POPULATION is NOT normally distributed the SDSM will look normal if n is large usually okay if n 30 or more So let s look at what happens to the SDSM with a non normal population Go back to the website above Click on Clear Lower 3 and change the population from Normal to Custom You can now click on the population graph and create your own population distribution Create a bimodal population Simulate sampling 1000 samples Note that SDSM for n 25 looks much close to normally distributed compared to the SDSM for n 5 Why do we care if the SDSM is normally distributed or not In future lectures we will talk about the assumption of normality for many statistical tests However at this point if the SDSM is normally distributed then we can calculate the probability of collecting a SAMPLE of a certain size from a population with known parameters Remember the z equation for calculating the probability of sample is And the z equation for calculating the probability of a sample means is 7 Individual scores of a placement exam are normally distributed with a mean of 84 2 and a standard deviation of 12 8 a If the score of one individual is randomly selected find the probability that the score will be more than 84 2pts a 0 508 b If a random sample of size n 20 is selected find the probability that the sample mean will be more than 84 2pts a 0 5279 8 Choose the correct term of the 2 given in parentheses to make each statement a true statement 2pts each a For a normally distributed population the distribution of sample means will OFTEN or ALWAYS be normally distributed b For a normally distributed population the spread of the distribution of sample means will be equal to the SAMPLE or POPULATION standard deviation divided by the square root of n c For populations that are not normally distributed the spread of the distribution of OBSERVATIONS or MEANS will approach normality when n is large