BINOMIAL DISTRIBUTIONS 1 9 The sample average random variable X In these trials the sample size n will be the same for all the samples The population mean is considered to be a fixed quantity but the sample mean x will differ from sample to sample Example randomly select samples of 4 persons each and calculate the average height for each group The sample averages will be different from each other and different from the population mean Sample Space of the random variable X Random Process POPULATION OF INDIVIDUAL DATA VALUES X SAMPLING DISTRIBUTION OF SAMPLE MEANS X mean X standard deviation X mean X standard deviation X Notation X and X Sampling distribution The distribution of all possible values of a specific statistic obtained from all possible simple random samples of the same size drawn from the same population Next Step What is the PDF of this random variable 2022 Radha Bose Florida State University Department of Statistics BINOMIAL DISTRIBUTIONS 2 9 The Central Limit Theorem If X1 X2 Xn are iid with finite mean and finite variance 2 then as n increases the distribution of X will approach a Normal distribution regardless of the shape of the original distribution Guidelines for application of the CLT SHAPE OF ORIGINAL POPULATION OF INDIVIDUAL VALUES X Non symmetric left skewed right skewed etc Symmetric uniform etc Normal Bell t etc Symmetric uniform etc Non symmetric left skewed right skewed etc SHAPE OF SAMPLING DISTRIBUTION OF X Normal Bell t etc 2022 Radha Bose Florida State University Department of Statistics BINOMIAL DISTRIBUTIONS 3 9 If we do not know the shape of the original population then to be on the safe side n should be 31 or more Regardless of the population shape and regardless of the sample size n the expected value mean of the sampling distribution of X will be the same as the mean of the original population X X the standard deviation of the sampling distribution of X will be X X n n the above formula shows that sample means vary less than individual values So the z score of a sample mean will be zx x n When n 1 the sampling distribution of X is the same as the distribution of the individual values X so it is the same as the original population distribution The Central Limit Theorem states that regardless of the population shape the sampling distribution of X will approach a Normal distribution as n increases That is the sampling distribution will start out looking like the original population n 1 and as the samples get bigger it will gradually morph into a bell shape and eventually assume the shape of a Normal distribution so that it can be modeled by a Normal random variable The sampling distribution of X is therefore considered to be N n under certain conditions see guidelines on previous page The random variable X is an estimator of the population mean because we often use our particular sample result x as an estimate of The random variable X is an unbiased estimator of the population mean because E X Side note E S2 2 so the sample variance random variable S2 is an unbiased estimator of the population variance 2 If we divided by n instead of n 1 when calculating s2 then the expected value would have been n 1 2 n and S2 would not have been an unbiased estimator of 2 Sample means have a tendency to gather around their mean the tendency gets stronger as n gets higher This implies that as n increases there will be more sample means 2022 Radha Bose Florida State University Department of Statistics BINOMIAL DISTRIBUTIONS 4 9 near the center of the sampling distribution where is located more area near the center of the curve and fewer sample means in the tails less area in the tails Since larger samples produce values of x that stay closer to it follows that a larger sample is more likely to give us a better estimate of than a smaller sample ILLUSTRATION Population is 5 10 15 50 Population mean is 5 10 15 50 4 20 n 1 n 2 n 3 n 4 n 5 n 6 55 50 45 40 35 30 25 20 15 10 5 0 Mean of sampling distributions mean of population Sampling distributions become more bell shaped as n increases Interpreting probabilities or percentages related to x You will need to cater these formats to suit the particular story and you may need to cater them to achieve proper grammar 2022 Radha Bose Florida State University Department of Statistics BINOMIAL DISTRIBUTIONS 5 9 There is a insert here the probability shaded on your picture probability that the average observation of n randomly selected subjects will be insert here a verbal description of the interval of values bolded on the data axis Insert here the percentage of data shaded on your picture of all possible random samples of n subjects each are expected to have average observations that are insert here a verbal description of the interval of values bolded on the data axis Example A Florida Highway Patrol officer has observed that the speeds of the vehicles on their stretch of the highway are uniformly distributed between 55 mph and 85 mph with mean 70 mph and standard deviation 8 7 mph Sub vehicles Obs speed Pop Dis symmetric a We wish to model the average speeds of all samples of n vehicles with a Normal distribution How many vehicles should we have in each sample in order for it to be safe to use a Normal model 15 or more b What is the approximate distribution of X the mean speed of 20 randomly chosen vehicles on the officer s stretch of highway Mean st deviation N 70 8 5 square root 20 x c The officer observed that a certain group of 20 vehicles had an average speed of 64 mph Should the officer be concerned that something might be wrong or is this just a typical average speed Z score x m st dev 64 7 0 8 7 square root 20 3 08 More then 3 sta dev away from mean so unsuiall d 2022 Radha Bose Florida State University Department of Statistics BINOMIAL DISTRIBUTIONS 6 9 e A certain group of 20 vehicles had an average speed of W mph and we know that ZW 1 9 How far was W from the mean in mph z standard deviation 1 9 8 7 square root 20 3 7 mph f A certain group of 20 vehicles had an average speed of W mph and we know that ZW 2 8 What was W Zw w m sd 2 8 w 70 sd 75 4 g Write out in words in context what the shaded area represents in the picture below X n 20 70 80 h What percent of all groups of 20 vehicles each are expected to have an average speed of 65 mph or more 99 5 2022 Radha Bose Florida State University Department of Statistics BINOMIAL DISTRIBUTIONS 7 9 i What percent of the z …