6.4 The Normal Approximation to the Binomial Distribution Recall from section 6.4 that g A binomial experiment is a experiment that satisfies the following four requirements: 1. Each trial can have only two outcomes, success and failure. 2. There must be a fixed number of trials. 3. The outcomes of each trial must be independent of each other. 4. The of a success must remain the same for each trial. g A binomial distribution is the distribution of a binomial experiment Notation for the Binomial Distribution p = P (S ) = the of success q = P (F ) = the of failure Note that q = 1 p n = the number of trials X = the number of successes Binomial Probability Formula In a binomial experiment, the of exactly X successes in n trials is P (X ) = nc X p X q n X = n! (n X)!X! px q n X Example 1: Use the binomial formula to find P (E ) where E = a student correctly guessing 4 of 10 multiple- choice questions (5 choices each question). S = correct answer p =.2 F = incorrect answer q =.8 P (E ) = 10! 1! 4! (.2 ( )4.8) 6 n = 10 X = 4 = 0.0881 Example 2: Use the binomial formula to find P (E ) where E = a student correctly guessing at most 2 of 10 multiple- choice questions (5 choices each question). P (E ) = P (0 correct) + P (1 correct) + P (2 correct) = 10! 10! 0! (.2 ( )0.8) 10 + 10! 9! 1! (.2 ( )1.8) 9 + 10! 8! 2! (.2 ( )2.8) 8 = (.8) 10 +10 (.2)(.8) 9 + 45 (.2) 2 (.8) 8 = 0.10737 + 0.26844 + 0.30200 = 0.6778 Suppose we were interested in the of a student correctly guessing at most 150 of 175 multiple- choice questions. This method would be tedious! Under certain conditions, a binomial distribution closely approximates the normal curve. Page 1 of 5
Let s complete the probabilities for a 10 question multiple- choice quiz P(0) = 0.10737 P(1) = 0.26844 P(2) = 0.30200 P(3) = P(4) = 0.08810 P(5) = P(6) = P(7) = P(8) = P(9) = P(10) = And then complete the distribution for this binomial experiment. Probability 0.3 0.2 0.1 l l l l l l l l l l 1 2 3 4 5 6 7 8 9 10 Number of trials (n) Notice the approximation of a normal bell curve. Page 2 of 5
When p = 0.5 and n 10, the binomial distribution will be approximately normal. Also, when np 5 and nq 5, the distribution will be approximately normal. Compare the symmetric and bell shape qualities of the following 6 distributions. When a binomial distribution is approximately normal, we can apply the techniques used for finding probabilities of normally distributed data. However, we must apply a correction for continuity as follows: Binomial Normal 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) Page 3 of 5
Example 3: If a baseball player s batting average is.320, find the that the player will get at most 26 hits in 100 times at bat. Step 0: Can the problem be approximated by a normal distribution? Step 1 Step 2 Step 3 Step 4 Step 5 Find µ and. µ = np = npq Write the problem in notation. P(X ) = Continuity correction factor: P(X > ) = Find the corresponding z-values: z 1 = X µ z 2 = X µ Label and shade. 3 2 1 0 1 2 3 Find the desired P(X > ) = P(z > ) µ 3 µ 2 µ µ µ+ µ+2 µ+3 Page 4 of 5
Example 4: Find P (E ) where E = a student correctly guessing exactly 8 of 20 multiple- choice questions (4 choices each question). Step 0: Can the problem be approximated by a normal distribution? Step 1 Step 2 Step 3 Step 4 Step 5 Find µ and. µ = np = npq Write the problem in notation. P(X = ) = Continuity correction factor: P( < X < ) = Find the corresponding z-values: z 1 = X µ z 2 = X µ Label and shade. 3 2 1 0 1 2 3 Find the desired P(7.5 < X < 8.5) = P( < z < ) = µ 3 µ 2 µ µ µ+ µ+2 µ+3 Homework: 6.4 p. 360 # 5 12 Page 5 of 5