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Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Lecture Slides Elementary Statistics Twelfth Edition and the Triola Statistics Series by Mario F. Triola
Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Chapter 6 Normal Probability Distributions 6-1 Review and Preview 6-2 The Standard Normal Distribution 6-3 Applications of Normal Distributions 6-4 Sampling Distributions and Estimators 6-5 The Central Limit Theorem 6-6 Assessing Normality 6-7 Normal as Approximation to Binomial
Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Key Concept This section presents a method for using a normal distribution as an approximation to the binomial probability distribution. If the conditions of np ≥ 5 and nq ≥ 5 are both satisfied, then probabilities from a binomial probability distribution can be approximated well by using a normal distribution with mean and standard deviation:
Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Review Binomial Probability Distribution 1. The procedure must have a fixed number of trials. 2. The trials must be independent. 3. Each trial must have all outcomes classified into two categories (commonly, success and failure). 4.The probability of success remains the same in all trials. Solve by binomial probability formula, Table A-1, or technology.
Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Approximation of a Binomial Distribution with a Normal Distribution then, and the random variable has distribution. (normal) a
Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Procedure for Using a Normal Distribution to Approximate a Binomial Distribution 1. Verify that both and. If not, you must use software, a calculator, a table or calculations using the binomial probability formula. 2.Find the values of the parameters μ and σ by calculating: 3. Identify the discrete whole number x that is relevant to the binomial probability problem. Focus on this value temporarily.
Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. 4.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 equal to x – 0.5, and mark the right side with the number equal to x Consider the entire area of the entire strip to represent the probability of the discrete whole number itself. 5.Determine whether the value of x itself is included in the probability. 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 or left of the strip; also shade the interior of the strip if and only if x itself is to be included. This total shaded region corresponds to the probability being sought. Procedure for Using a Normal Distribution to Approximate a Binomial Distribution
Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. 6.Using x – 0.5 or x in place of x, find the area of the shaded region: find the z score, use that z score to find the area to the left of the adjusted value of x. Use that cumulative area to identify the shaded area corresponding to the desired probability. Procedure for Using a Normal Distribution to Approximate a Binomial Distribution
Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. In 431 NFL football games that went to over time, the teams that won the coin toss went on to win 235 of those games. If the coin-toss method is fair, teams winning the toss would win about 50% of the games (we’d expect wins in 431 overtime games). Assuming there is a 0.5 probability of winning a game after winning the coin toss, find the probability of getting at least 235 winning games. Example – NFL Coin Toss
Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. The given problem involves a binomial distribution with n = 431 trials and an assumed probability of success of p = 0.5. Use the normal approximation to the binomial distribution. Step 1: The conditions check: Example – NFL Coin Toss
Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Step 2: Find the mean and standard deviation of the normal distribution: Step 3: We want the probability of at least 235 wins, so x = 235. Step 4: The vertical strip will go from to Example – NFL Coin Toss
Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Step 5: We will shade the area to the right of Example – NFL Coin Toss
Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Step 6: Find the z score and use technology or Table A-2 to determine the probability. The probability is for the coin flip winning team to win at least 235 games. This probability is low enough to suggest the team winning coin flip has an unfair advantage. Example – NFL Coin Toss
Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Definition When we use the normal distribution (which is a continuous probability distribution) as an approximation to the binomial distribution (which is discrete), a continuity correction is made to a discrete whole number x in the binomial distribution by representing the discrete whole number x by the interval from x – 0.5 to x (that is, adding and subtracting 0.5).
Section Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example – Continuity Corrections
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