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# Chapter 15: Sampling Distributions MATH 243

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This 18 page Class Notes was uploaded by Rachel Kasashima on Monday October 19, 2015. The Class Notes belongs to MATH 243 at University of Oregon taught by Harker H in Fall 2015. Since its upload, it has received 29 views. For similar materials see Intro Probability and Statistics in Mathematics (M) at University of Oregon.

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Date Created: 10/19/15

Chapter lS Sampling Distributions The primary question is Whether we can believe the results of a random sam ple or a randomized comparative eX periment Is the element of chance affecting the data we collect De nitions parameter is a number that describes the population It is usually not known statistic is a number that describes a sample statistic often estimates an unknown parameter Pa m meter Statistic M 6 09gtlt Example We want to measure the mean amount spent purchasing houses in Oregon This is the parameter We interview 500 peo ple Who purchased houses and nd the mean purchase price for this sample of house buyers This is our statistic How well does our statistic approximate our parameter De nition The population distribution of a vari able is the distribution of values of the variable among individuals 11 house purchase prices would give a population distribution Not something You 60quot WWW qat De nition The sampling distribution of a statistic is the distribution of values taken by the statistic in all possible samples of the same size from the same popula tlon The means of all samples of size 500 give a sampling distribution How many samples of a Particular 5126 are in the population17 quot n nPopuuaHon r wk r gamma Size Problem The length of human pregnancies varies according to a distribution that is ap proximately normal With mean 266 days and standard deviation l6 daysWhat is the probability that the preg nancy length for 6 randomly chosen women exceeds 270 days DONOT MIX NUMBERS up I Know Popula on dust is appmx normal N7b6 lb Wb Sampling distribution for 3 is N we 4 Mn 027 r O Z39bb 23259 26b 46 Law of Large Numbers s the sample size increases the mean X of the sample gets closer to the mean of the population In practice it may be too expensive to get a large sample Consider the set of numbers sample space S 2357 8 The mean is 5 Take all samples of size 1 These Will be the numbers themselves The mean is 5 The standard deviation is 25495 NOW take all samples of size 2 There are 10 2939 25 27 28 35 37 38 57 58 78 The means for these samples are 25 35 45 5 4 5 55 6 65 75 The mean of the sample means is 5 The standard deviation of the sample means is 14720 Mum OF the de means S I avs we same We popula on mom HOWEVER Standard deviation almmeple means decreases The probability distribution of these means is an example of a sampling distribu tion There are 9 possible outcomes X 2535445 5566575 mam saw a a X are closer 11 the Imam IN GENERAL Suppose that X is the mean of an SRS of size n Suppose our population has mean and standard deviation 6 Then 0 The mean of the sampling distribu tion is o The standard deviation of the sam pling distribution is oIf our population follows 6 then G W the sampling distribution follows What does this mean o The statistic X is an unbiased estima tor of o verages X s are less variable than individual observations 0 The results of large samples are less variable than the results of small sam ples The Central Limit Theorem The sampling distribution of means of random samples of size n from a popu lation With mean and standard deVi ation G is approximately 6 7 When n is large Note The population distribution does not need to be normal Note that knowing the standard deVi ation of the sampling distribution and the fact that it is approximately Nor mal we know how likely it is that a sample is within that deviation from the mean Hence how far our sample mean is from from real mean Example Homeowners loss to re has mean 250 and 6 1000 The distribution is rightskewed If a company sells 10000 policies can it safely base its rates on the assumption that its average loss Will be no greater than 275 Yew 4397 Mame 39000 NZ90r W lt samplc dlc rnlaAlIon kl0 mm M z zquot 140 250 2b0 Ono Leo WV likely that ne averaqe 05 Win be 5 11 Example The number of accidents Ber week at a hazardous intersection varies With mean M 22 and standard deviation 146 1 What is the approximate distribution Of 7 OVGI weeks sample distribution 7 Folww approx Nzz L 2 What is the approximate probability that E lt 2 m 2 2139 22 2i391 3 What is the approximate probability that there are fewer than 100 acci dents in a year auidenfs 00 P691 szweeks lt 3900 K lt a Z lfx is distributed by NM6 i then 2 is asmma wN m io e samplim dist bum

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