1、统计学基础(英文版第7版)课件les7e_ppt_ADA_0505Chapter Outline 5.1Introduction to Normal Distributions and the Standard Normal Distribution 5.2Normal Distributions:Finding Probabilities 5.3Normal Distributions:Finding Values 5.4Sampling Distributions and the Central Limit Theorem 5.5Normal Approximations to Binom
2、ial DistributionsSection 5.5Normal Approximations to Binomial DistributionsSection 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 probabilitiesNorm
3、al 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 success in a single tr
4、ial,and q is the probability of failure in a single trial.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.Example:Approximating a Binomial Distribution(1 of 2)Two binomial experiments are lis
5、ted.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 deviation.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 p
6、oorgrades(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:Kaiser Family Foundation)Example:Approximating a Binomial Distribution(2 of 2)2.In a survey of 8-to 18-year-old light media users in
7、 the United States,23said they get fair or poorgrades(C and below).You randomly select twenty 8-to 18-year-old light media users in the United States and ask them whether they get fair or poor grades.(Source:Kaiser Family Foundation)Solution:Approximating the Binomial Distribution(1 of 2)1.In this b
8、inomial experiment,45n,0.47p,and 0.53q.So,45 0.4721.15np and 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.Solution:Approximating the Binomial distribution(2 of 2)1.In the figure,notice tha
9、t the binomial distribution is approximately bell-shaped,which supports the conclusion that you can use a normal distribution to approximate the distribution of x.Solution:Approximating the Binomial2.In this binomial experiment,20n,0.23p,and 0.77q.So,20 0.234.6np and 20 0.7715.4nq.Because 5np,you ca
10、nnot 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.Correction for Continuity(1 of 2)A binomial distribution is discrete and ca
11、n 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 adding the areas of bars in the probability histogram.Correction for Continuity(2 of 2)When you use a con
12、tinuous normal distribution to approximate a binomial probability,you need to move 0.5 unit to the left and right of the midpoint to include all possible x-values in the interval(continuity correction).Example:Using a Continuity Correction(1 of 3)Use a continuity correction to convert each binomial
13、probability to a normal distribution probability.1.The probability of getting between 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 269.5310.5
14、Px.Example:Using a Continuity Correction(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 continuo
15、us normal distribution is 157.5x.The normaldistribution probability is 157.5P x.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 successes.Solution:The discret
16、e midpoint values are,60,61,62.The corresponding interval for the continuousnormal distribution is 62.5x.The normaldistribution probability is 62.5P x.Binomial Probabilities Involving the Number cUsing a Normal Distribution to Approximate Binomial ProbabilitiesIn WordsIn Symbols1.Verify that the bin
17、omial 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 andstandard deviation for thedistribution.npnpqUsing the Normal Distribution to Approximate Binomial ProbabilitiesIn WordsIn Symbols4.A
18、pply the appropriate continuity correction.Shade the corresponding area under the normal curve.Add 0.5 to(or subtract 0.5 from)the binomial probability.5.Find the corresponding z-score(s).xz6.Find the probability.Use the Standard Normal Table.Example:Approximating a Binomial Probability(1 of 3)In a
19、survey of 8-to 18-year-old heavy media users in the United 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
20、respond yes?(Source:Kaiser Family Foundation)Solution:Use the normal approximation 21.15and 3.35 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 subtra
21、cting 0.5 from 20 and write the probability as 200.519.5P xP x.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 is xz19.521.150.493.35z.Using the Standard Normal Table,0
22、.490.3121P z .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.Example:Approximating a Binomial Probability(2 of 3)A study on aggressive driving found that of driverssay they have yelled
23、at another driver.You randomly select 200 drivers in the United States and ask them whether they have yelled at another driver.What is the probability that at least 100 drivers will say yes,they have yelled at another driver?(Source:American Automobile Association)Solution:Approximating a Binomial P
24、robability(4 of 10)Solution:Because 200 0.4794np and 200 0.53106nq,the binomial 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.Solution:Approximatin
25、g a Binomial Probability(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.Solution:Approximating a Binomial Probability(6 of 10)So,the probability that at least 100 drivers will say“ye
26、s”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 .Example:Approximating a Binomial Probability(3 of 3)A study of National Football League(NFL)retirees,ages 50 and older,found that have arthritis.You
27、randomly 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 Research)Solution:Approximating a Binomial Probability(7 of 10)SolutionBecause 75 0.62446.8n
28、p and 75 0.37628.2nq,the binomial variable x is approximately normally distributed,with 46.8npand 75 0.6240.3764.19npq.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 figu
29、re shows anormal curve with 46.8,4.19,and the shadedarea under the curve between 47.5 and 48.5.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.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.
