Normal Distribution as an Approximation to the Binomial Distribution

Transcription

1 Chapter 1 Student Lecture Notes 1-1 Normal Distribution as an Approximation to the Binomial Distribution : Goals ONE TWO THREE 2 Review Binomial Probability Distribution applies to a discrete random variable 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. solve by P(x) formula, computer software, or Table in the Annex

2 Chapter 1 Student Lecture Notes 1-2 The Normal Approximation to the Binomial Using the normal distribution (a continuous distribution) as a substitute for a binomial distribution (a discrete distribution) for large values of n seems reasonable because as n increases, a binomial distribution gets closer and closer to a normal distribution. The normal probability distribution is generally deemed a good approximation to the binomial probability distribution when nπ and n(1- π ) are both greater than 5. 4 The Normal Approximation continued Recall for the binomial experiment: There are only two mutually exclusive outcomes (success or failure) on each trial. A binomial distribution results from counting the number of successes. Each trial is independent. The probability is fixed from trial to trial, and the number of trials n is also fixed. 5 Approximate a Binomial Distribution with a Normal Distribution if: 1. np 5 2. nq 5

3 Chapter 1 Student Lecture Notes 1-3 Approximate a Binomial Distribution with a Normal Distribution if: 1. np 5 2. nq 5 Then µ = np and σ = npq and the random variable has a (normal) distribution. Binomial Distribution for an n of 3 and 20, where π =. n=3 n=20 P(x) number of occurences P(x) n u m b e r o f o c c u r e n c e s 8 Solving Binomial Probability Problems Using a Normal Approximation

4 Chapter 1 Student Lecture Notes 1-4 Solving Binomial Probability Problems Using a Normal Approximation Start After verifying that we have a binomial probability problem, identify n, p, q Is Computer Software Available? Use the Computer Software No Can the problem be solved by using Table A-1? No Can the problem be easily solved with the binomial probability formula? Use the Table A-1 Use binomial probability formula n! P(x) = p x q (n x)!x! n x Solving Binomial Probability Problems Using a Normal Approximation Can the problem be easily solved with the binomial probability formula? Use binomial probability formula n! P(x) = p x (n x)!x! q n x No Are np 5 and nq 5 both true? No Compute µ = np and σ = 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) 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 z = x µ σ where µ and σ 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.

5 Chapter 1 Student Lecture Notes 1-5 Normal Approximation to the Binomial: Continuity Correction Factor The value.5 subtracted or added, depending on the problem, to a selected value when a binomial probability distribution (a discrete probability distribution) is being approximated by a continuous probability distribution (the normal distribution). 13 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 For x =, draw a strip from to.5. Consider the area of the strip to represent the probability of discrete number x. continued continued Continuity Corrections Procedures 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.

6 Chapter 1 Student Lecture Notes 1-6 x = at least =, 65, 66,.... x = at least =, 65, 66,... x = more than = 65, 66, 67, x = at least =, 65, 66,... x = more than = 65, 66, 67,... x = at most = 0, 1,... 62, 63,

7 Chapter 1 Student Lecture Notes 1-7 x = at least =, 65, 66,... x = more than = 65, 66, 67,... x = at most = 0, 1,... 62, 63, x = fewer than = 0, 1,... 62, x = exactly x = exactly

8 Chapter 1 Student Lecture Notes 1-8 x = exactly.5 Interval represents discrete number Normal Approximation to the Binomial Example: Video camera A recent study by a marketing research firm showed that 15% of Canadian households owned a video camera. A sample of 200 homes is obtained. Of the 200 homes sampled how many would you expect to have video cameras? µ = n π = (. 15)( 200) = Normal Approximation to the Binomial Example: Video camera What is the variance? 2 σ = n π ( 1 π ) = ( 30)( 1. 15) = What is the standard deviation? σ = 255. = What is the probability that less than 40 homes in the sample have video cameras? We need P(X<40)=P(X<39). So, using the normal approximation, P(X<39.5) P[Z ( )/5.0498] = P(Z ) P(Z<1.88)=

9 Chapter 1 Student Lecture Notes 1-9 r a l i t r b u i o n : µ = 0, σ Normal Approximation to the Binomial 2 = 1 Example: Video camera f ( x Z= P(Z=1.88) = = Summary 26