The Distributions of the Sample Mean and Sample Proportion 1 Sample Means 2 2 Sample Proportions 6 3 Central Limit Theorem 9 www apsu edu jonesmatt 1530 htm 1 1 Sample Means It is reasonable to estimate the population mean with the sample mean x However before drawing a sample from a population the sample mean X is a random variable so it has a distribution Knowing the distribution of the sample mean X helps us to know how well X can be used to approximate www apsu edu jonesmatt 1530 htm 2 Example 1 The heights in inches of five starting players on a men s basketball team are Alfred 76 Bob 79 Carl 85 Dennis 82 Edgar 78 How many samples of two players can be taken List all possible samples and the sample means www apsu edu jonesmatt 1530 htm 3 The mean of the sample means is The mean height is This is not a coincidence The mean of all the sample means is always the mean of the population The population standard deviation of the players heights is The standard deviation of the mean heights is These numbers are different www apsu edu jonesmatt 1530 htm 4 Sample Mean Statistics The mean of the sample means is equal to the population mean x If sampling from an infinite population the standard deviation of the sample mean which is often called the standard error is x n The above can be shown with A LOT of algebra we ll skip the details The larger the sample size n the smaller the standard error tends to be in estimating by x www apsu edu jonesmatt 1530 htm 5 2 Sample Proportions It is reasonable to estimate the population proportion p with the sample proportion p of successes x n sample size n However before drawing a sample from a population the sample proportion p is a random variable so it has a distribution Knowing the distribution of the sample proportion p helps us to know how well p can be used to approximate p www apsu edu jonesmatt 1530 htm 6 Sample Proportions GUESS WHAT Proportions are means Example 2 In a survey 55 of 150 fans favored Sammy Hagar over David Lee Roth Of course the sample proportion is 55 p 150 Another way to see this is to assign a 1 to each vote for Sammy and a 0 to each vote for Roth Then 55 1s 95 0s z z 1 1 1 0 0 0 55 p 150 150 Then a proportion is a mean of 1s and 0s and so the results for the distribution of the sample mean also apply to proportions www apsu edu jonesmatt 1530 htm 7 Sample Proportion Statistics Because proportions are means the following results flow from the results on slide 5 Since p is a kind of x and p is a kind of p p Since p p 1 p is the std dev for one trial for a binomial experiment p n p p 1 p n r p 1 p n Here as with the case for means p is often called the standard error Note the larger the sample size n the smaller the standard error is when estimating p by p www apsu edu jonesmatt 1530 htm 8 3 Central Limit Theorem This is one of the most famous and important results in all of mathematics It forms the basis for many statistical tests Theorem 3 As the sample size n approaches the random variable X becomes distributed more and more like a normal with mean and standard deviation n This result is true for every population distribution with finite standard deviation www apsu edu jonesmatt 1530 htm 9 So What Does the Central Limit Theorem Imply The ages of pennies tend to have a right skewed distribution We d like to estimate the mean age of a penny in circulation Using Minitab compare the distributions of sample means for samples of size 5 and size 40 www apsu edu jonesmatt 1530 htm 10 Examples Example 4 The package states a pudding cup contains 99g of pudding Actually the amount of pudding in a cup has mean 101g and standard deviation 1g If you take a sample of 25 cups what s the approximate probability the sample mean amount of pudding is less than 100 5g What s approximate probability it will be between 100 6g and 101 4g Example 5 About 72 of all Halloween candy is undesirable If you take 50 pieces at random what is the probability that the undesirable proportion in your sample is between 70 and 74 What s the chance the undesirable proportion of candy in a sample of size 80 is between 70 and 74 www apsu edu jonesmatt 1530 htm 11