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1 Bluman, Chapter 6 1

2 Review the following from Chapter 5 A surgical procedure has an 85% chance of success and a doctor performs the procedure on 10 patients, find the following: a) The probability that the procedure was successful on exactly two patients? b) The mean, variance and standard deviation of number of successful surgeries. Bluman, Chapter 6 2

3 Approximating a Binomial Distribution But what if the doctor performs the surgical procedure on 150 patients and you want to find the probability of fewer than 100 successful surgeries? To do this using the techniques described in chapter 5, you would have to use the binomial formula 100 times and find the sum of the resulting probabilities. This is not practical and a better approach is to use a normal distribution to approximate the binomial distribution.

4 6.4 The Normal Approximation to the Binomial Distribution A normal distribution is often used to solve problems that involve the binomial distribution since when n is large (say, 100), the calculations are too difficult to do by hand using the binomial distribution. Bluman, Chapter 6 4

5 The Normal Approximation to the Binomial Distribution The normal approximation to the binomial is appropriate when np > 5 and nq > 5. I. μ = np II. σ = npq where q = 1 p See page 274 for above formulas. Bluman, Chapter 6 5

6 The Normal Approximation to the Binomial Distribution In addition, a correction for continuity may be used in the normal approximation to the binomial. The continuity correction means that for any specific value of X, say 8, the boundaries of X in the binomial distribution (in this case, 7.5 to 8.5) must be used. Bluman, Chapter 6 6

7 Correction for Continuity

8 Websites The following websites simulate the concept. /stat_sim/normal_approx/index.html omdist/binomdist.html Bluman, Chapter 6 8

9 The Normal Approximation to the Binomial Distribution; see page 342 Binomial Normal When finding: Use: P(X = a) P(a 0.5 < X < a + 0.5) P(X a) P(X > a 0.5) P(X > a) P(X > a + 0.5) P(X a) P(X < a + 0.5) P(X < a) P(X < a 0.5) For all cases, np, npq, np 5, nq 5 Bluman, Chapter 6 9

10 Bluman, Chapter 6 10

11 The Normal Approximation to the Binomial Distribution Procedure Table Step 1: Check to see whether the normal approximation can be used. Step 2: Find the mean µ and the standard deviation. Step 3: Write the problem in probability notation, using X. Step 4: Rewrite the problem by using the continuity correction factor, and show the corresponding area under the normal distribution. Step 5: Find the corresponding z values. Step 6: Find the solution. Bluman, Chapter 6 11

12 Ex. 3: Approximating a Binomial Probability Thirty-seven percent of Americans say they always fly an American flag on the Fourth of July. You randomly select 15 Americans and ask each if he or she flies an American flag on the Fourth of July. What is the probability that fewer than eight of them reply yes? SOLUTION: From Example 1, you know that you can use a normal distribution with = 5.55 and 1.87 to approximate the binomial distribution. By applying the continuity correction, you can rewrite the discrete probability P(x < 8) as P (x < 7.5). The graph on the next slide shows a normal curve with = 5.55 and 1.87 and a shaded area to the left of 7.5. The z-score that corresponds to x = 7.5 is

13 Continued... z x Using the Standard Normal Table, P (z<1.04) = So, the probability that fewer than eight people respond yes is

14 Ex. 4: Approximating a Binomial Probability Twenty-nine percent of Americans say they are confident that passenger trips to the moon will occur during their lifetime. You randomly select 200 Americans and ask if he or she thinks passenger trips to the moon will occur in his or her lifetime. What is the probability that at least 50 will say yes? SOLUTION: Because np = = 58 and nq = = 142, the binomial variable x is approximately normally distributed with np 58 and npq

15 Ex. 4 Continued Using the correction for continuity, you can rewrite the discrete probability P (x 50) as the continuous probability P ( x 49.5). The graph shows a normal curve with = 58 and = 6.42, and a shaded area to the right of 49.5.

16 Ex. 4 Continued The z-score that corresponds to 49.5 is z x So, the probability that at least 50 will say yes is: P(x 49.5) = 1 P(z -1.32) = =

17 Chapter 6 Normal Distributions Section 6-4 Example 6-16 Page #343 Bluman, Chapter 6 17

18 Example 6-16: Reading While Driving A magazine reported that 6% of American drivers read the newspaper while driving. If 300 drivers are selected at random, find the probability that exactly 25 say they read the newspaper while driving. Here, p = 0.06, q = 0.94, and n = 300. Step 1: Check to see whether a normal approximation can be used. np = (300)(0.06) = 18 and nq = (300)(0.94) = 282 Since np 5 and nq 5, we can use the normal distribution. Step 2: Find the mean and standard deviation. µ = np = (300)(0.06) = npq Bluman, Chapter 6 18

19 Example 6-16: Reading While Driving Step 3: Write in probability notation. Step 4: Rewrite using the continuity correction factor. P(24.5 < X < 25.5) Step 5: Find the corresponding z values z 1.58, z Step 6: Find the solution P(X = 25) The area between the two z values is = , or 2.27%. Hence, the probability that exactly 25 people read the newspaper while driving is 2.27%. Bluman, Chapter 6 19

20 Chapter 6 Normal Distributions Section 6-4 Example 6-17 Page #343 Bluman, Chapter 6 20

21 Example 6-17: Widowed Bowlers Of the members of a bowling league, 10% are widowed. If 200 bowling league members are selected at random, find the probability that 10 or more will be widowed. Here, p = 0.10, q = 0.90, and n = 200. Step 1: Check to see whether a normal approximation can be used. np = (200)(0.10) = 20 and nq = (200)(0.90) = 180 Since np 5 and nq 5, we can use the normal distribution. Step 2: Find the mean and standard deviation. µ = np = (200)(0.06) = npq Bluman, Chapter 6 21

22 Example 6-17: Widowed Bowlers Step 3: Write in probability notation. Step 4: Rewrite using the continuity correction factor. P(X > 9.5) Step 5: Find the corresponding z values z Step 6: Find the solution P(X 10) The area to the right of the z value is = , or 99.34%. The probability of 10 or more widowed people in a random sample of 200 bowling league members is 99.34%. Bluman, Chapter 6 22

23 Study Tip In a discrete distribution, there is a difference between P (x c) and P( x > c). This is true because the probability that x is exactly c is not zero. In a continuous distribution, however, there is no difference between P (x c) and P (x >c) because the probability that x is exactly c is zero.

24 homework Sec 6.4 page 346 #5-13 odds Bluman, Chapter 6 24

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