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1、Elementary Statistics,Seventh Edition,Chapter 5,Normal Probability Distributions,Copyright 2019, 2015, 2012, Pearson Education, Inc.,Chapter Outline,5.1Introduction to Normal Distributions and the Standard Normal Distribution5.2Normal Distributions: Finding Probabilities5.3Normal Distributions: Find
2、ing Values5.4Sampling Distributions and the Central Limit Theorem5.5Normal Approximations to Binomial Distributions,Section 5.5,Normal Approximations to Binomial Distributions,Section 5.5 Objectives,How to determine when the normal distribution can approximate the binomial distributionHow to find th
3、e continuity correctionHow to use the normal distribution to approximate binomial probabilities,Normal Approximation to a Binomial (1 of 2),Normal Approximation to a Binomial Distribution,If,and, then the binomial random,variable x is approximately normally distributed with,mean,standard deviation,w
4、here n is the number of independent trials, p is the probability of success in a single trial, and q is the probability of failure in a single trial.,Normal Approximation to a Binomial (2 of 2),Binomial distribution:, and,and,.,As n increases the histogram approaches a normal curve.,Example: Approxi
5、mating 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 deviation. If you cannot, explain why.,In a survey of 8- t
6、o 18-year-old heavy media users in the United States,said they get fair or poor,grades (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
7、 Binomial Distribution (2 of 2),In a survey of 8- to 18-year-old light media users in the United States,said they get fair or poor,grades (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: Kai
8、ser Family Foundation),Solution: Approximating the Binomial Distribution (1 of 2),In this binomial experiment, and,. So,and,.,Because np and nq are greater than 5, you can use a normal distribution with,and,to approximate the distribution of x.,Solution: Approximating the Binomial distribution (2 of
9、 2),In the figure, notice that 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 Binomial,In this binomial experiment, and,. So,and,.,Because, you cannot use a
10、 normal distribution to,approximate 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 can be
11、 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 co
12、ntinuous 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 bi
13、nomial probability to a normal distribution probability.,The probability of getting between 270 and 310 successes, inclusive.,Solution:The discrete midpoint values are 270, 271, , 310. The corresponding interval for the continuous,normal distribution is,. The normal,distribution probability is,.,Exa
14、mple: Using a Continuity Correction (2 of 3),Use a continuity correction to convert each binomial probability to a normal distribution probability.,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,. The normal,distribution probability is,.,Example: Using a Continuity Correction (3 of 3),Use a continuity correction to convert each binomial probability to a normal distribution probability.,The probability of getting fewer than 63 successes.,Solution:The discrete midpoi
16、nt values are ,60, 61, 62.The corresponding interval for the continuous,normal distribution is,. The normal,distribution probability is,.,Binomial Probabilities Involving the Number c,Using a Normal Distribution to Approximate Binomial Probabilities,In Words,In Symbols,Verify that the binomial distr
17、ibution applies.,Specify n, p, and q.,Determine if you can use the normal distribution to approximate x, the binomial variable.,Is,?,Is,?,Find the mean,and,standard deviation,for the,distribution.,Using the Normal Distribution to Approximate Binomial Probabilities,In Words,In Symbols,Apply the appro
18、priate continuity correction. Shade the corresponding area under the normal curve.,Add 0.5 to (or subtract 0.5 from) the binomial probability.,Find the corresponding z-score(s).,Find the probability.,Use the Standard Normal Table.,Example: Approximating a Binomial Probability (1 of 3),In a survey of
19、 8- to 18-year-old heavy media users in the United States,said they get fair or poor,grades (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 res
20、pond yes?(Source: Kaiser Family Foundation),Solution:Use the normal approximation,and,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 subtracti
21、ng 0.5 from 20 and write the probability as,.,Solution: Approximating a Binomial Probability (2 of 10),The figure shows a normal curve with, and the shaded area to the left of 19.5.,The z-score that corresponds to,is,.,Using the Standard Normal Table,.,Solution: Approximating a Binomial Probability
22、(3 of 10),The probability that fewer than twenty 8- to 18-year-olds respond yes is approximately 0.3121, or about,.,Example: Approximating a Binomial Probability (2 of 3),A study on aggressive driving found that,of drivers,say they have yelled at another driver. You randomly select 200 drivers in th
23、e 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 Probability (4 of 10),Solution:Because,and,the bin
24、omial variable x is approximately normally distributed, with,and,.,Using the continuity correction, you can rewrite the discrete probability,as the continuous,probability,.,Solution: Approximating a Binomial Probability (5 of 10),Solution:The figure shows a normal,curve with,and the shaded area to t
25、he right of 99.5.,The z-score that corresponds to 99.5 is,.,Solution: Approximating a Binomial Probability (6 of 10),So, the probability that at least 100 drivers will say “yes” is approximately,.,The probability that at least 100 drivers will say “yes” is approximately 0.2177, or about,.,Example: A
26、pproximating a Binomial Probability (3 of 3),A study of National Football League (NFL) retirees, ages 50 and older, found that,have arthritis. You,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
27、yes? (Source: University of Michigan, Institute for Social Research),Solution: Approximating a Binomial Probability (7 of 10),SolutionBecause,and,the binomial variable x is approximately normally distributed, with,and,.,Solution: Approximating a Binomial Probability (8 of 10),Using the continuity co
28、rrection, you can rewrite the discrete probability,as the continuous,probability,. The figure shows a,normal curve with, and the shaded,area 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,and the z-score that corresponds to 48.5 is,.,Solution: Approximating a Binomial Probability (10 of 10),So, the probability that exactly 48 NFL retirees will say they have arthritis is,.,The probability that exactly 48 NFL retirees will say they have arthritis is approximately 0.0916, or about,.,
