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Section 6-6 Normal as Approximation to Binomial Remember Binomial Probability Distribution 1. There must be a fixed number of trials. 2. Each trial must be independent. 3. Every trial has exactly two possible outcomes. 4. The probability of each outcome remains consistent throughout every trial. New Notes for Binomials When both ππ β₯ 5 and ππ β₯ 5, then a binomial can be treated as a normal distribution with the mean π = ππ and π = πππ. Remember: β’ p is probability of success β’ q is probability of failure (1-p) β’ n is number of trials Continuity Correction β’ X represents the number of successes which is discrete, however normal distributions are for continuous random variables. β’ As a result, we must do a βcontinuity correctionβ, in which we create a 1 unit interval around x, by considering x - 0.5 and x + 0.5 The How To For x < 120: Calculate the area to the left of 119.5 For x < 120: Calculate the area to the left of 120.5 The How To For x > 120: Calculate the area to the right of 120.5 For x > 120: Calculate the area to the right of 119.5 The How To For x = 120: Calculate the area between 119.5 and 120.5 Overall Process β’ Given Binomial: 1. Determine if it can be approximated as a normal by checking if np and nq are > 5. 2. Find π = ππ and π = πππ 3. Identify the value of X and determine the appropriate continuity correction 4. Calculate the z Score for the value(s) found in step 3 (using π and π from step 2). 5. Use table to find probability Example 1 β’ If a gender-selection technique is tested with 500 couples that have 1 baby, find the probability of getting at least 275 girls. Solution 1. This can be approximated using normal because ππ = ππ = 500(.5) = 250 β₯ 5. 2. π = ππ = 500 .5 = 250, π = πππ = 500 .5 (.5) = 125 = 11.18 3. Since βat leastβ means β₯ we want to find the area to the right of 275 β 0.5 = 274.5 4. π§ = π₯βπ π = 274.5β250 11.18 = 2.19 Solution (Continued) 5. The area to the left of π§ = 2.19 is .9857, so the area to the right is π΄ = 1 β .9857 = .0143 6. The conclusion is that the probability of 273 female births out of 500 is .0143. Example 2 β’ Suppose that a sample of 20 tires of the same type are obtained at random. It is understood that 8% of all the tires are defective. What is the probability that 15 or fewer of the collected tires are defective? Example 3 β’ If 10% of men are bald, what is the probability of selecting less than 100 bald men in a sample of 818 men. Example 4 β’ 62% of 12th graders living in Oswego attend OHS. If a sample of 500 12th graders from the city are selected, find the probability that more than 290 of them go to OHS. Example 5 β’ If you flip a coin 20 times, what is the probability of it landing on heads exactly 10 times? Homework Pg. 306-307 #5-9,14-16, 21(a, b), 27 Homework Answers Pg. 306-307 #5-9,14-16, 21(a, b), 27 5. The area to the right of 8.5. 6. The area to the right of 1.5. 7. The area to the left of 4.5. 8. The area between 3.5 and 4.5. 9. The area to the left of 15.5. 14. normal approx. is not suitable 15. normal approx. is not suitable 16. Normal approx: 0.0329 21. a.) 0.0318 b.) 0.2676 27. 0.2776. No, 27 blue M&Ms is not unusually high because 0.2776 is not small. HOMEWORK QUIZ: β’ 62% of 12th graders living in Oswego attend OHS. If a sample of 500 12th graders from the city are selected, find the probability that more than 290 of them go to OHS.