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Review of Statistical Terms • • • • Population Sample Parameter Statistic Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 1 Population the set of all measurements (either existing or conceptual) under consideration Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 2 Sample a subset of measurements from a population Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 3 Statistic a numerical descriptive measure of a sample Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 4 Parameter a numerical descriptive measure of a population Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 5 We use a statistic to make inferences about a population parameter. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 6 Principal types of inferences • Estimate the value of a population parameter • Formulate a decision about the value of a population parameter • Make a prediction about the value of a statistical variable. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 7 Sampling Distribution a probability distribution for the sample statistic we are using Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 8 Example of a Sampling Distribution Select samples with two elements each (in sequence with replacement) from the set {1, 2, 3, 4, 5, 6}. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 9 Constructing a Sampling Distribution of the Mean for Samples of Size n = 2 List all samples and compute the mean of each sample. sample: mean: sample: mean {1,1} {1,2} {1,3} {1,4} {1,5} 1.0 1.5 2.0 2.5 3.0 {1,6} {2,1} {2,2} … 3.5 1.5 4 ... There are 36 different samples. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 10 Sampling Distribution of the Mean Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . x p 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36 11 Sampling Distribution Histogram 6 36 1 36 | 1 | | | | | | | | | | | 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 12 Let x be a random variable with a normal distribution with mean and standard deviation . Let x be the sample mean corresponding to random samples of size n taken from the distribution. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 13 Facts about sampling distribution of the mean: Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 14 Facts about sampling distribution of the mean: • The x distribution is a normal distribution. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 15 Facts about sampling distribution of the mean: • The x distribution is a normal distribution. • The mean of the x distribution is (the same mean as the original distribution). Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 16 Facts about sampling distribution of the mean: • The x distribution is a normal distribution. • The mean of the x distribution is (the same mean as the original distribution). • The standard deviation of the x distribution is n (the standard deviation of the original distribution, divided by the square root of the sample size). Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 17 We can use this theorem to draw conclusions about means of samples taken from normal distributions. If the original distribution is normal, then the sampling distribution will be normal. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 18 The Mean of the Sampling Distribution x Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 19 The mean of the sampling distribution is equal to the mean of the original distribution. x Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 20 The Standard Deviation of the Sampling Distribution x Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 21 The standard deviation of the sampling distribution is equal to the standard deviation of the original distribution divided by the square root of the sample size. x n Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 22 The time it takes to drive between cities A and B is normally distributed with a mean of 14 minutes and a standard deviation of 2.2 minutes. • Find the probability that a trip between the cities takes more than 15 minutes. • Find the probability that mean time of nine trips between the cities is more than 15 minutes. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 23 Mean = 14 minutes, standard deviation = 2.2 minutes • Find the probability that a trip between the cities takes more than 15 minutes. 15 14 z 0.45 14 15 2.2 P( z 0.45) 1.00 0.6736 0.3264 Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . Find this area 24 Mean = 14 minutes, standard deviation = 2.2 minutes • Find the probability that mean time of nine trips between the cities is more than 15 minutes. x 14 2.2 x 0.73 n 9 Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 25 Mean = 14 minutes, standard deviation = 2.2 minutes • Find the probability that mean time of nine trips between the cities is more than 15 minutes. Find this area 15 14 z 1.37 0.73 14 15 P( z 1.37 ) 0.5 0.4147 0.0853 Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 26 What if the Original Distribution Is Not Normal? Use the Central Limit Theorem. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 27 Central Limit Theorem If x has any distribution with mean and standard deviation , then the sample mean based on a random sample of size n will have a distribution that approaches the normal distribution (with mean and standard deviation divided by the square root of n) as n increases without bound. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 28 How large should the sample size be to permit the application of the Central Limit Theorem? In most cases a sample size of n = 30 or more assures that the distribution will be approximately normal and the theorem will apply. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 29 Central Limit Theorem • For most x distributions, if we use a sample size of 30 or larger, the x distribution will be approximately normal. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 30 Central Limit Theorem • The mean of the sampling distribution is the same as the mean of the original distribution. • The standard deviation of the sampling distribution is equal to the standard deviation of the original distribution divided by the square root of the sample size. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 31 Central Limit Theorem Formula x Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 32 Central Limit Theorem Formula x n Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 33 Central Limit Theorem Formula z x x x x / n Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 34 Application of the Central Limit Theorem Records indicate that the packages shipped by a certain trucking company have a mean weight of 510 pounds and a standard deviation of 90 pounds. One hundred packages are being shipped today. What is the probability that their mean weight will be: a. b. c. more than 530 pounds? less than 500 pounds? between 495 and 515 pounds? Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 35 Are we authorized to use the Normal Distribution? Yes, we are attempting to draw conclusions about means of large samples. Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 36 Applying the Central Limit Theorem What is the probability that their mean weight will be more than 530 pounds? Consider the distribution of sample means: x 510 , x 90 / 100 9 P( x > 530): z = 530 – 510 = 20 = 2.22 9 9 .0132 P(z > 2.22) = _______ Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 37 Applying the Central Limit Theorem What is the probability that their mean weight will be less than 500 pounds? P( x < 500): z = 500 – 510 = –10 = – 1.11 9 9 .1335 P(z < – 1.11) = _______ Copyright (C) 2006 Houghton Mifflin Company. All rights reserved . 38