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This 12 page Class Notes was uploaded by Cale Steuber on Friday October 2, 2015. The Class Notes belongs to STT 1810 at Appalachian State University taught by Ross Gosky in Fall. Since its upload, it has received 64 views. For similar materials see /class/217698/stt-1810-appalachian-state-university in Statistics at Appalachian State University.
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Date Created: 10/02/15
Chapter 9 Sampling Distributions The concept of a sampling distribution is difficult to understand at rst but it is an important statistical concept Motivating Example Suppose we wanted to know the probabilities of the following events How would we nd them Event of Interest How we d nd the probability It rains on a July Day in Boone Height of an adult male is above 70 inches A voter favors increasing the local sales tax The average height of a group of 10 adult males is above 70 inches A sample of 200 voters has a majority that favor the sales tax increase The last two examples are examples of sampling distributions STT1810 Course Notes ChaDters 9 and 10 Page 1 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 population Example Sampling Distribution of the proportion of heads in 20 ips of a fair coin STT1810 Course Notes Chapters 9 and 10 Page 2 Two Key Results Sampling Distribution Result 1 The Sampling Distribution of the Sample Mean Let X be a random variable from M distribution with mean u and standard deviation 5 The 039 distribution of X will approximately follow a N 1 distribution as the sample size increases Rule of Thumb The number of observations required to apply the central limit theorem does vary somewhat depending on the distribution of X In general though for this class we will assume a sample size of n30 is suf cient for the result to apply What does this result tell us 7 1 From this result we can conclude m follows a N0l distribution the standard normal distribution Example to Show Validi of Result 2 STT1810 Course Notes Chapters 9 and 10 Page 3 Example 1 Let X represent a man s height Suppose that X follows a N68 4 distribution What is the probability that one randomly selected man will be over 70 inches tall Now suppose we observe 30 randomly selected values of X What is the probability that the average of these 30 heights will be larger than 70 inches Example 2 Flight prices from Charlotte to Boston average 525 with standard deviation of 75 A business traveler will y this route 50 times in the neXt year Supposing each of these 50 ights is a random selection from the population of all CLTBOS ights what is the probability her average cost of the ight is above 550 STT1810 Course Notes Chapters 9 and 10 Page 4 Sampling Distribution Result 2 Let f7 represent a sample proportion and p represent a A 100 p population proportion Forlarge n p N N p 7 at least approx1mately n This approximation tends to be reasonable when np and nl p are both at least 10 Example 1 Suppose that 54 of the eligible voters in a city will vote for a candidate on Election Day Before the election a newspaper is interested in estimating the candidate s support so they take a poll of 300 random voters in the city a Let f7 represent the proportion of voters in a poll of 300 voters that support the candidate What is the approximate sampling distribution of f7 b Interpret the result in part A in words C Find the probability that the newspaper s poll shows a minority that support the candidate STT1810 Course Notes Chapters 9 and 10 Page 5 Example 2 Survey sample calls in New York city only have a l in 12 chance of succeeding Suppose we plan to make 500 calls to randomly selected numbers in NYC a Let f7 represent the proportion of calls that are answered when 500 calls are made What is the approximate sampling distribution of f7 b Interpret the result in part A in words C What is the probability that more than 10 of a set of 500 calls made by a pollster in NYC lead to successful responses STT1810 Course Notes Chapters 9 and 10 Page 6 Chapter 10 Con dence Intervals for Population Proportions The idea of a con dence interval is to draw a conclusion about a population parameter from a sample statistic For Stt1810 we focus on the following situation Population Parameter p a proportion of interest unknown Sample Statistic f7 a sample proportion calculated from a random sample of size n from the population of interest A 1 We will use the key result of p N Np n to form our con dence intervals for p based upon the sample proportion Recall that the empirical rule tells us that 95 of the time any Normally distributed variable is within 2 standard deviations of its mean A more technical answer from the Z table suggests that 95 of the time any Normally distributed variable is within 196 standard deviations of its mean How can we use this result here What are our conclusions STT1810 Course Notes ChaDters 9 and 10 Page 7 Can we repeat these calculations with a different probability level like say 90 of 99 What would change in our previous argument We will use these general ndings to produce what we call con dence intervals for u The general form of these intervals is estimate margin of error These intervals also have a con dence level which is a measurement of certainty that the interval actually contains the parameter of interest STT1810 Course Notes Chapters 9 and 10 Page 8 Con dence Intervals for a Po ulation Pro ortion Con dence Intervals for p A t A l A p i Z M where 2 depends upon the con dence level des1red 71 Con dence Level 2 90 1 645 95 196 99 2576 These intervals tend to be reasonably accurate Whenever there are at least 10 successes in the sample data and at least 10 failures in the sample data and as long as the population is at least 10 times as large as the sample size Example A large state university with 15000 students is contemplating switching from the quarter system to a semester system The administration conducts a survey of a random sample of 400 students and nds that 240 of them prefer to remain on the quarter system a Construct s 90 CI for the true proportion of all students who prefer to remain on the quarter system b Construct a 95 CI for the same quantity 0 Construct a 99 CI for the same quantity 1 What do we notice about the margin of error as the con dence level increases STT1810 Course Notes Chapters 9 and 10 Page 9 e Do these calculations provide convincing evidence that more than half of all students prefer to remain on the quarter system Explain f Do we believe this CI formula is reasonably accurate in this case Why or why not Example 2 Example 1063 in the book shows an interesting example of question wording Gallup used two different questions to assess support for the death penalty in a February 1999 poll Question 1 Are you in favor of the death penalty for a person convicted of murder 11 543 for 71 against 22 no opinion 7 Question 2 What do you think should be the penalty for murder 7 the death penalty or life imprisonment with absolutely no possibility of parole n 511 death penalty 56 life imprisonment 38 no opinion 6 Compute 95 C1s for the true proportion of adults who favor the death penalty in each case Question 1 Question 2 STT1810 Course Notes Chapters 9 and 10 Page 10 How Con dence Intervals Behave The ideal scenario for a con dence interval would be high con dence AND a small margin of error hence an interval with small width Unfortunately achieving both these objectives is difficult Often researchers are forced to choose between one or the other Let s focus on the margin of error The margin of error in our con dence intervals for the mean A l A 1s Z M n A small margin of error is always desirable How can this margin of error be small Wavs the margin of error can be small 1 2 can be made small This is equivalent to choosing a smaller con dence level C for the interval So we can get a smaller interval by sacri cing some of our desired con dence in the correctness of the answer N V The sample proportion phat can be close to 0 or 1 It turns out that plp is small when p is closer to 0 or 1 and this reduces the margin of error However we can t control this as researchers So our leverage in controlling this factor is Virtually nonexistent W V The sample size 11 could be large A large sample size does decrease the margin of error And the sample size is something that is within the control of the researcher This is one reason why we ve said that larger sample sizes are better STT1810 Course Notes Chapters 9 and 10 Page 11 Choosin the Sam 1e Size for a Desired Mar in of Error The con dence interval for a population mean will have a speci ed margin of error B when the 2 sample size is n f7l if A 1 A This can be veri ed by noting that the margin of error B Z M and then solV1ng this 7 equation for n the sample size Note 1 Because f7 is unknown when this calculation is taking place do the following 1 2 Note 2 often when you solve for the needed sample size you will not get a round number for your answer In that case Example 1 A poll via Videocamera will be taken to determine the rate that drivers run red lights in a particularly busy intersection in a large city How large a sample size is needed so that the margin of error for a 95 CI for the true rate is a 5 or less b 2 or less Note in the above calculation we assumed that f7 would be STT1810 Course Notes Chapters 9 and 10 Page 12
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