SECTION 5.5 Normal Approximations to Binomial Distributions 277 Continuity Correction A binomial distribution is discrete and can be represented by a probability histogram. To calculate exact binomial probabilities, you can use the binomial formula for each value of x and add the results. Geometrically, this corresponds to adding the areas of bars in the probability histogram (see top figure at the left). Remember that each bar has a width of one unit and x is the midpoint of the interval. 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 (see bottom figure at the left). When you do this, you are making a continuity correction. Using a Continuity Correction Use a continuity correction to convert each binomial probability to a normal distribution probability. 1. The probability of getting between 270 and 310 successes, inclusive 2. The probability of getting at least 158 successes 3. The probability of getting fewer than 63 successes SOLUTION 1. The discrete midpoint values are 270, 271, . . ., 310. The corresponding interval for the continuous normal distribution is 269.5 6 x 6 310.5 and the normal distribution probability is P1269.5 6 x 6 310.52. 2. The discrete midpoint values are 158, 159, 160, . . .. The corresponding interval for the continuous normal distribution is x 7 157.5 and the normal distribution probability is P1x 7 157.52. 3. The discrete midpoint values are . . ., 60, 61, 62. The corresponding interval for the continuous normal distribution is x 6 62.5 and the normal distribution probability is P1x 6 62.52. TRY IT YOURSELF 2 Use a continuity correction to convert each binomial probability to a normal distribution probability. 1. The probability of getting between 57 and 83 successes, inclusive 2. The probability of getting at most 54 successes Answer: Page A39 Shown below are several cases of binomial probabilities involving the number c and how to convert each to a normal distribution probability. Binomial Normal Notes Exactly c P1c - 0.5 6 x 6 c + 0.52 Includes c At most c P1x 6 c + 0.52 Includes c Fewer than c P1x 6 c - 0.52 Does not include c At least c P1x 7 c - 0.52 Includes c More than c P1x 7 c + 0.52 Does not include c EXAMPLE 2 c P(x = c) P(c − 0.5 < x < c + 0.5) c + 0.5 c − 0.5 x c x Normal approximation Exact binomial probability Study Tip In a discrete distribution, there is a difference between P1x Ú c2 and P1x 7 c2. This is true because the probability that x is exactly c is not 0. In a continuous distribution, however, there is no difference between P1x Ú c2 and P1x 7 c2 because the probability that x is exactly c is 0.
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