Transcript Ch7-Sec7.4
HAWKES LEARNING SYSTEMS math courseware specialists Copyright © 2008 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Section 7.4 Approximating the Binomial Distribution Using the Normal Distribution HAWKES LEARNING SYSTEMS Sampling Distributions math courseware specialists 7.4 Approx. the Binomial Dist. Using the Normal Dist. Review of Binomial Distribution: • The experiment consists of n identical trials. • Each trial is independent of the others. • For each trial, there are only two possible outcomes. For counting purposes, one outcome is labeled a success, the other a failure. • For every trial, the probability of getting a success is called p. The probability of getting a failure is then 1 – p. • The binomial random variable, X, is the number of successes in n trials. HAWKES LEARNING SYSTEMS Sampling Distributions math courseware specialists 7.4 Approx. the Binomial Dist. Using the Normal Dist. Normal Distribution Approximation of a Binomial Distribution: If the conditions that np ≥ 5 and n(1 – p) ≥ 5 are met for a given binomial distribution, then a normal distribution can be used to approximate its probability distribution with the given mean and standard deviation: HAWKES LEARNING SYSTEMS Sampling Distributions math courseware specialists 7.4 Approx. the Binomial Dist. Using the Normal Dist. Continuity Correction: A continuity correction is a correction factor employed when using a continuous distribution to approximate a discrete distribution. Examples of the Continuity Correction Statement Symbolically Area At least 45, or no less than 45 ≥ 45 Area to the right of 44.5 More than 45, or greater than 45 > 45 Area to the right of 45.5 At most 45, or no more than 45 ≤ 45 Area to the left of 45.5 Less than 45, or fewer than 45 < 45 Area to the left of 44.5 Exactly 45, or equal to 45 = 45 Area between 44.5 and 45.5 HAWKES LEARNING SYSTEMS Sampling Distributions math courseware specialists 7.4 Approx. the Binomial Dist. Using the Normal Dist. Calculate the probability: Use the continuity correction factor to describe the area under the normal curve that approximates the probability that at least 2 people, in a statistics class of 50, cheated on the last test. Assume that the number of people who cheated is a binomial distribution with a mean of 5 and a standard deviation of 2.12. Solution: Begin by adding and subtracting 0.5 to and from 2. Draw a normal curve indicating the interval 1.5 to 2.5 to represent 2. Next, shade the area corresponding to the phrase at least 2. HAWKES LEARNING SYSTEMS Sampling Distributions math courseware specialists 7.4 Approx. the Binomial Dist. Using the Normal Dist. Process for Using the Normal Curve to Approximate the Binomial Distribution: 1. Determine the values of n and p. 2. Verify that the conditions np ≥ 5 and n(1 – p) ≥ 5. 3. Calculate the values of the mean and standard deviation using the formulas and . 4. Use a continuity correction to determine the interval corresponding to the value of x. 5. Draw a normal curve labeled with the information in the problem. 6. Convert the value of the random variable(s) to a z-value(s). 7. Use the normal curve table to find the appropriate area under the curve. HAWKES LEARNING SYSTEMS Sampling Distributions math courseware specialists 7.4 Approx. the Binomial Dist. Using the Normal Dist. Calculate the probability: After many hours of studying for your statistics test, you believe that you have a 90% probability of answering any given question correctly. Your test included 50 true/false questions. What is the probability that you will miss no more than 4 questions? Solution: n = 50, p = 0.10 since we are looking at questions missed. np = 5 and n(1 – p) = 45, both which are greater than or equal to 5. = 50(0.10) =5 2.121 HAWKES LEARNING SYSTEMS Sampling Distributions math courseware specialists 7.4 Approx. the Binomial Dist. Using the Normal Dist. Solution (continued): Use the continuity correction by adding and subtracting 0.5 to and from 4. Draw a normal curve indicating the interval 3.5 to 4.5 to represent 4. - 0.24 P(z ≤ - 0.24) = 0.4052 HAWKES LEARNING SYSTEMS Sampling Distributions math courseware specialists 7.4 Approx. the Binomial Dist. Using the Normal Dist. Calculate the probability: Many toothpaste commercials advertise that 3 out of 4 dentists recommend their brand of toothpaste. What is the probability that out of a random survey of 400 dentists, 300 will have recommended Brand X toothpaste? Assume that the commercials are correct, and therefore, there is a 75% chance that any given dentist will recommend Brand X toothpaste. Solution: n = 400, p = 0.75 np = 300 and n(1 – p) = 100, both which are greater than or equal to 5. = 400(0.75) = 300 8.660 HAWKES LEARNING SYSTEMS Sampling Distributions math courseware specialists 7.4 Approx. the Binomial Dist. Using the Normal Dist. Solution (continued): Use the continuity correction by adding and subtracting 0.5 to and from 300. Draw a normal curve indicating the interval 299.5 to 300.5 to represent 300. - 0.06 and P(-0.06 ≤ z ≤ 0.06) = 0.0478 0.06