Normal Distribution as an Approximation to the Binomial

Review Binomial Probability Distribution v applies to a discrete random variable v has these requirements: 1. The experiment must have fixed number of trials. 2. The trials must be independent. 3. Each trial must have all outcomes classified into two categories. 4. The probabilities must remain constant for each trial. v solve by P(x) formula, computer software, or Table A-1 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 2

Approximate a Binomial Distribution with a Normal Distribution if: 1. np ³ 5 2. nq ³ 5 Then µ = np and s = npq and the random variable has a (normal) distribution. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 4

Figure 5 -24 Solving Binomial Probability Problems Using a Normal Approximation Copyright © 1998,

Figure 5 -24 Solving Binomial Probability Problems Using a Normal Approximation Start 1 2 3 After verifying that we have a binomial probability problem, identify n, p, q 4 Is Computer Software Available ? Yes Use the Computer Software No Can the problem be solved by using Table A-1 ? Yes Use the Table A-1 No Can the problem be easily solved with the binomial probability formula ? Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman Yes Use binomial probability formula n! P(x) = (n – x)!x! • p x • q n–x 6

Figure 5 -24 Solving Binomial Probability Problems Using a Normal Approximation Can the problem be easily solved with the binomial probability formula ? 7 4 5 6 Yes Use binomial probability formula n! P(x) = (n – x)!x! • p x • q n–x No Are np ³ 5 and nq ³ 5 both true ? No Yes Compute µ = np and s =Ö npq Draw the normal curve, and identify the region representing the probability to be found. Be sure to include the continuity correction. (Remember, the discrete value x is adjusted for continuity by adding and subtracting 0. 5) Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 7

Figure 5 -24 Solving Binomial Probability Problems Using a Normal Approximation Draw the normal curve, and identify the region representing the probability to be found. Be sure to include the continuity correction. (Remember, the discrete value x is adjusted for continuity by adding and subtracting 0. 5) Calculate 7 8 9 z =x – µ s where µ and s are the values already found and x is adjusted for continuity. Refer to Table A-2 to find the area between µ and the value of x adjusted for continuity. Use that area to determine the probability being sought. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 8

Continuity Corrections Procedures 1. When using the normal distribution as an approximation to the binomial distribution, always use the continuity correction. 2. In using the continuity correction, first identify the discrete whole number x that is relevant to the binomial probability problem. 3. Draw a normal distribution centered about µ, then draw a vertical strip area centered over x. Mark the left side of the strip with the number x - 0. 5, and mark the right side with x + 0. 5. For x = 64, draw a strip from 63. 5 to 64. 5. Consider the area of the strip to represent the probability of discrete number x. continued Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 9

Continuity Corrections Procedures continued 4. Now determine whether the value of x itself should be included in the probability you want. Next, determine whether you want the probability of at least x, at most x, more than x, fewer than x, or exactly x. Shade the area to the right of left of the strip, as appropriate; also shade the interior of the strip itself if and only if x itself is to be included, The total shaded region corresponds to probability being sought. Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 10

x = at least 64 . = 64, 65, 66, . . . 50

x = at least 64 . = 64, 65, 66, . . . 50 64 63. 5 x = more than 64 = 65, 66, 67, . . . 50 x = at most 64 = 0, 1, . . . 62, 63, 64 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 65 64. 5 50 64 64. 5 13

x = at least 64 . = 64, 65, 66, . . . 50 64 63. 5 x = more than 64 = 65, 66, 67, . . . 50 x = at most 64 = 0, 1, . . . 62, 63, 64 65 64. 5 50 64 64. 5 x = fewer than 64 = 0, 1, . . . 62, 63 50 63 63. 5 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 14

x = exactly 64 Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 15

x = exactly 64 50 64 Copyright © 1998, Triola, Elementary Statistics Addison Wesley

x = exactly 64 50 63. 5 64. 5 Interval represents discrete number 64

Chapter 5 Normal Probability Distributions 5 -1 Overview 5 -2 The Standard Normal Distribution 5 -3 & 5 -4 Nonstandard Normal Distributions (Finding Probabilities & Finding Scores) 5 -5 The Central Limit Theorem 5 -6 Normal Distributions as Approximation to Binomial Distribution Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 18

Basic Concepts • Continuous distribution/Density curve • Uniform distribution • Normal distribution – Standard normal distribution • Central Limit Theorem (Approx. normal distr. ) – Distribution of sample mean • mean, variance, standard deviation (standard error) – finite population correction factor – continuity correction (Binomial distribution) Copyright © 1998, Triola, Elementary Statistics Addison Wesley Longman 19