Section 5.5 Normal Approximations to Binomial Distributions Larson/Farber 4th ed

Section 5.5 Objectives Determine when the normal distribution can approximate the binomial distribution Find the correction for continuity Use the normal distribution to approximate binomial probabilities Larson/Farber 4th ed

Normal Approximation to a Binomial Normal Approximation to a Binomial Distribution If np  5 and nq  5, then the binomial random variable x is approximately normally distributed with  mean μ = np  standard deviation The normal distribution is used to approximate the binomial distribution when it would be impractical to use the binomial distribution to find a probability. Larson/Farber 4th ed

Normal Approximation to a Binomial Binomial distribution: p = 0.25 As n increases the histogram approaches a normal curve. Larson/Farber 4th ed

1.(#6, pg 291) A survey of U.S. adults found that 63% would want their organs transplanted into a patient who needs them if they were killed in an accident. You randomly select 20 adults and ask each if he or she would want their organs transplanted into a patient who needs them if they were killed in an accident. Example: Approximating the Binomial Decide whether you can use the normal distribution to approximate x, the number of people who reply yes. If you can, find the mean and standard deviation. Larson/Farber 4th ed

Solution: Approximating the Binomial You can use the normal approximation n = 20, p = 0.63, q = 0.37 np = (20)(0.63) = ≥ 5 nq = (20)(0.37) = 7.40 ≥ 5 Mean: μ = np = Standard Deviation:

2.(#8, pg 291) A survey of workers in the United States found that 8.6% work fewer than 40 hours per week. You randomly select 30 workers in the United States and ask each if he or she work fewer than 40 hours per week. Example: Approximating the Binomial Decide whether you can use the normal distribution to approximate x, the number of people who reply yes. If you can find, find the mean and standard deviation. Larson/Farber 4th ed

Solution: Approximating the Binomial You cannot use the normal approximation n = 30, p = 0.086, q = np = (30)(0.086) = 2.58 < 5 nq = (30)(0.914) = ≥ 5 Because np < 5, you cannot use the normal distribution to approximate the distribution of x. Larson/Farber 4th ed

Correction for Continuity The binomial distribution is discrete and can be represented by a probability histogram. To calculate exact binomial probabilities, the binomial formula is used for each value of x and the results are added. Geometrically this corresponds to adding the areas of bars in the probability histogram. Larson/Farber 4th ed

Correction for Continuity When you use a continuous normal distribution to approximate a binomial probability, you need to move 0.5 unit to the left and right of the midpoint to include all possible x-values in the interval (correction for continuity). Exact binomial probability P(x = c) c Normal approximation P(c – 0.5 < x < c + 0.5) cc+ 0.5c– 0.5 Larson/Farber 4th ed

Example: Using a Correction for Continuity Use a correction for continuity to convert the binomial intervals to a normal distribution interval. 1.The probability of getting between 27 and 31 successes, inclusive. Solution: The discrete midpoint values are 27, 27, …, 31. The corresponding interval for the continuous normal distribution is 26.5 < x < 31.5 Larson/Farber 4th ed

Example: Using a Correction for Continuity Use a correction for continuity to convert the binomial intervals to a normal distribution interval. 2.The probability of getting at least 15 successes. Solution: The discrete midpoint values are 15, 16, 17, …. The corresponding interval for the continuous normal distribution is x > 14.5 Larson/Farber 4th ed

Example: Using a Correction for Continuity Use a correction for continuity to convert the binomial intervals to a normal distribution interval. 3.The probability of getting less than 15 successes. Solution: The discrete midpoint values are …,12, 13, 14. The corresponding interval for the continuous normal distribution is x < 14.5 Larson/Farber 4th ed

Using the Normal Distribution to Approximate Binomial Probabilities Verify that the binomial distribution applies. 1.Determine if you can use the normal distribution to approximate x, the binomial variable. 3.Find the mean  and standard deviation  for the distribution. Is np  5? Is nq  5? Specify n, p, and q. Larson/Farber 4th ed In Words In Symbols

Using the Normal Distribution to Approximate Binomial Probabilities 4.Apply the appropriate continuity correction. Shade the corresponding area under the normal curve. 5.Find the corresponding z- score(s). 6.Find the probability. Add or subtract 0.5 from endpoints. Use the Standard Normal Table. Larson/Farber 4th ed In Words In Symbols

Example: Approximating a Binomial Probability (#20, pg 292) Thirty-four percent of people in the United States have type A + blood. You randomly select 32 people in the United States and ask them if their blood type is A +. Solution: n = 32, p = 0.34, q = 0.66 np = (32)(0.34) = ≥ 5 nq = (32)(0.66) = ≥ 5 μ = np = (32)(0.34) = 10.88

Solution: Approximating a Binomial Probability Apply the continuity correction: Exactly 12 people say they have A + blood corresponds to the continuous normal distribution interval 11.5 < x < 12.5 Normal Distribution μ = σ = 2.68 a.P(x = 12) = P(11.5 < x < 12.5) = P(0.23 < z < 0.60) = = b.P(x ≥ 12) = P(x > 11.5) = P(z > 0.23) = = c.P(x < 12) = P(x < 11.5) = P(z < 0.23) =

Solution: Approximating a Binomial Probability P(x < 60) = P(x < 59.5) = P(z < 1.47) = d.n = 150, np = 51 & nq = 99

Section 5.5 Summary Determined when the normal distribution can approximate the binomial distribution Found the correction for continuity Used the normal distribution to approximate binomial probabilities HW: EO