1、 Slide 1Elementary StatisticsSeventh EditionChapter 5Normal Probability DistributionsCopyright 2019,2015,2012,Pearson Education,Inc.Slide 2Copyright 2019,2015,2012,Pearson Education,Inc.Chapter Outline 5.1 Introduction to Normal Distributions and the Standard Normal Distribution 5.2 Normal Distribut
2、ions:Finding Probabilities 5.3 Normal Distributions:Finding Values 5.4 Sampling Distributions and the Central Limit Theorem 5.5 Normal Approximations to Binomial Distributions Slide 3Copyright 2019,2015,2012,Pearson Education,Inc.Section 5.5Normal Approximations to Binomial Distributions Slide 4Copy
3、right 2019,2015,2012,Pearson Education,Inc.Section 5.5 Objectives How to determine when the normal distribution can approximate the binomial distribution How to find the continuity correction How to use the normal distribution to approximate binomial probabilities Slide 5Copyright 2019,2015,2012,Pea
4、rson Education,Inc.Normal Approximation to a Binomial(1 of 2)Normal Approximation to a Binomial Distribution If 5np and 5nq,then the binomial randomvariable x is approximately normally distributed with mean np standard deviation npq where n is the number of independent trials,p is the probability of
5、 success in a single trial,and q is the probability of failure in a single trial.Slide 6Copyright 2019,2015,2012,Pearson Education,Inc.Normal Approximation to a Binomial(2 of 2)Binomial distribution:0.25p,1 0.25q ,and 4n,10n,25n and 50n.As n increases the histogram approaches a normal curve.Slide 7C
6、opyright 2019,2015,2012,Pearson Education,Inc.Example:Approximating a Binomial Distribution(1 of 2)Two binomial experiments are listed.Determine whether you can use a normal distribution to approximate the distribution of x,the number of people who reply yes.If you can,find the mean and standard dev
7、iation.If you cannot,explain why.1.In a survey of 8-to 18-year-old heavy media users in the United States,47said they get fair or poorgrades(C and below).You randomly select forty-five 8-to 18-year-old heavy media users in the United States and ask them whether they get fair or poor grades.(Source:K
8、aiser Family Foundation)Slide 8Copyright 2019,2015,2012,Pearson Education,Inc.Example:Approximating a Binomial Distribution(2 of 2)2.In a survey of 8-to 18-year-old light media users in the United States,23said they get fair or poorgrades(C and below).You randomly select twenty 8-to 18-year-old ligh
9、t media users in the United States and ask them whether they get fair or poor grades.(Source:Kaiser Family Foundation)Slide 9Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Approximating the Binomial Distribution(1 of 2)1.In this binomial experiment,45n,0.47p,and 0.53q.So,45 0.4721.15np and
10、45 0.5323.85nq.Because np and nq are greater than 5,you can use a normal distribution with 21.15npand 45 0.470.533.35npqto approximate the distribution of x.Slide 10Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Approximating the Binomial distribution(2 of 2)1.In the figure,notice that the
11、binomial distribution is approximately bell-shaped,which supports the conclusion that you can use a normal distribution to approximate the distribution of x.Slide 11Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Approximating the Binomial2.In this binomial experiment,20n,0.23p,and 0.77q.So,
12、20 0.234.6np and 20 0.7715.4nq.Because 5np,you cannot use a normal distribution toapproximate the distribution of x.Notice the binomial distribution is skewed right,which supports the conclusion that you cannot use a normal distribution to approximate the distribution of x.Slide 12Copyright 2019,201
13、5,2012,Pearson Education,Inc.Correction for Continuity(1 of 2)A binomial distribution is discrete and can be represented by a probability histogram.To calculate exact binomial probabilities,the binomial formula is used for each value of x and the results are added.Geometrically this corresponds to a
14、dding the areas of bars in the probability histogram.Slide 13Copyright 2019,2015,2012,Pearson Education,Inc.Correction for Continuity(2 of 2)When you use a continuous normal distribution to approximate a binomial probability,you need to move 0.5 unit to the left and right of the midpoint to include
15、all possible x-values in the interval(continuity correction).Slide 14Copyright 2019,2015,2012,Pearson Education,Inc.Example:Using a Continuity Correction(1 of 3)Use a continuity correction to convert each binomial probability to a normal distribution probability.1.The probability of getting between
16、270 and 310 successes,inclusive.Solution:The discrete midpoint values are 270,271,310.The corresponding interval for the continuousnormal distribution is 269.5310.5x.The normaldistribution probability is 269.5310.5Px.Slide 15Copyright 2019,2015,2012,Pearson Education,Inc.Example:Using a Continuity C
17、orrection(2 of 3)Use a continuity correction to convert each binomial probability to a normal distribution probability.2.The probability of getting at least 158 successes.Solution:The discrete midpoint values are 158,159,160,.The corresponding interval for the continuous normal distribution is 157.5
18、x.The normaldistribution probability is 157.5P x.Slide 16Copyright 2019,2015,2012,Pearson Education,Inc.Example:Using a Continuity Correction(3 of 3)Use a continuity correction to convert each binomial probability to a normal distribution probability.3.The probability of getting fewer than 63 succes
19、ses.Solution:The discrete midpoint values are,60,61,62.The corresponding interval for the continuousnormal distribution is 62.5x.The normaldistribution probability is 62.5P x.Slide 17Copyright 2019,2015,2012,Pearson Education,Inc.Binomial Probabilities Involving the Number c Slide 18Copyright 2019,2
20、015,2012,Pearson Education,Inc.Using a Normal Distribution to Approximate Binomial ProbabilitiesIn WordsIn Symbols1.Verify that the binomial distribution applies.Specify n,p,and q.2.Determine if you can use the normal distribution to approximate x,the binomial variable.Is 5np?Is 5nq?3.Find the mean
21、andstandard deviation for thedistribution.npnpq Slide 19Copyright 2019,2015,2012,Pearson Education,Inc.Using the Normal Distribution to Approximate Binomial ProbabilitiesIn WordsIn Symbols4.Apply the appropriate continuity correction.Shade the corresponding area under the normal curve.Add 0.5 to(or
22、subtract 0.5 from)the binomial probability.5.Find the corresponding z-score(s).xz6.Find the probability.Use the Standard Normal Table.Slide 20Copyright 2019,2015,2012,Pearson Education,Inc.Example:Approximating a Binomial Probability(1 of 3)In a survey of 8-to 18-year-old heavy media users in the Un
23、ited States,said they get fair or poorgrades(C and below).You randomly select forty-five 8-to 18-year-old heavy media users in the United States and ask them whether they get fair or poor grades.What is the probability that fewer than 20 of them respond yes?(Source:Kaiser Family Foundation)Solution:
24、Use the normal approximation 21.15and 3.35 Slide 21Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Approximating a Binomial Probability(1 of 10)Apply the continuity correction:To use a normal distribution,note that the probability is“fewer than 20.”So,apply the continuity correction by subtr
25、acting 0.5 from 20 and write the probability as 200.519.5P xP x.Slide 22Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Approximating a Binomial Probability(2 of 10)The figure shows a normal curve with 21.15,3.35,and the shaded area to the left of 19.5.The z-score that corresponds to 19.5x i
26、s xz19.521.150.493.35z.Using the Standard Normal Table,0.490.3121P z .Slide 23Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Approximating a Binomial Probability(3 of 10)The probability that fewer than twenty 8-to 18-year-olds respond yes is approximately 0.3121,or about 31.21.Slide 24Copyr
27、ight 2019,2015,2012,Pearson Education,Inc.Example:Approximating a Binomial Probability(2 of 3)A study on aggressive driving found that of driverssay they have yelled at another driver.You randomly select 200 drivers in the United States and ask them whether they have yelled at another driver.What is
28、 the probability that at least 100 drivers will say yes,they have yelled at another driver?(Source:American Automobile Association)Slide 25Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Approximating a Binomial Probability(4 of 10)Solution:Because 200 0.4794np and 200 0.53106nq,the binomial
29、 variable x is approximately normally distributed,with 94npand 200 0.470.537.06npq.Using the continuity correction,you can rewrite the discrete probability 100P x as the continuousprobability 99.5P x.Slide 26Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Approximating a Binomial Probability
30、(5 of 10)Solution:The figure shows a normalcurve with 94,7.06,and the shaded area to the right of 99.5.The z-score that corresponds to 99.5 is99.5940.78200 0.470.53z.Slide 27Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Approximating a Binomial Probability(6 of 10)So,the probability that a
31、t least 100 drivers will say“yes”is approximately 99.50.78P xP z10.78P z 1 0.7823 0.2177.The probability that at least 100 drivers will say“yes”is approximately 0.2177,or about 21 8 .Slide 28Copyright 2019,2015,2012,Pearson Education,Inc.Example:Approximating a Binomial Probability(3 of 3)A study of
32、 National Football League(NFL)retirees,ages 50 and older,found that have arthritis.Yourandomly select 75 NFL retirees who are at least 50 years old and ask them whether they have arthritis.What is the probability that exactly 48 will say yes?(Source:University of Michigan,Institute for Social Resear
33、ch)Slide 29Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Approximating a Binomial Probability(7 of 10)SolutionBecause 75 0.62446.8np and 75 0.37628.2nq,the binomial variable x is approximately normally distributed,with 46.8npand 75 0.6240.3764.19npq.Slide 30Copyright 2019,2015,2012,Pearson
34、 Education,Inc.Solution:Approximating a Binomial Probability(8 of 10)Using the continuity correction,you can rewrite the discrete probability 48P x as the continuousprobability 47.548.5Px.The figure shows anormal curve with 46.8,4.19,and the shadedarea under the curve between 47.5 and 48.5.Slide 31C
35、opyright 2019,2015,2012,Pearson Education,Inc.Solution:Approximating a Binomial Probability(9 of 10)The z-score that corresponds to 47.5 is 147.546.80.1775 0.6240.376zand the z-score that corresponds to 48.5 is 248.546.80.4175 0.6240.376z.Slide 32Copyright 2019,2015,2012,Pearson Education,Inc.Solution:Approximating a Binomial Probability(10 of 10)So,the probability that exactly 48 NFL retirees will say they have arthritis is 47.548.50.170.41PxPz0.410.17P zP z0.6591 0.56750.0916.The probability that exactly 48 NFL retirees will say they have arthritis is approximately 0.0916,or about 9.
