SECTION 5.2 Normal Distributions: Finding Probabilities 247 μ = 41 x Time (in minutes) 10 20 30 40 50 60 70 80 Finding Probabilities for Normal Distributions A survey indicates that for each trip to a supermarket, a shopper spends an average of 41 minutes with a standard deviation of 12 minutes in the store. The lengths of time spent in the store are normally distributed and are represented by the variable x. A shopper enters the store. (a) Find the probability that the shopper will be in the store for each interval of time listed below. (b) When 200 shoppers enter the store, how many shoppers would you expect to be in the store for each interval of time listed below? (Adapted from Time Use Institute) 1. Between 20 and 50 minutes 2. More than 35 minutes SOLUTION 1. (a) The figure at the left shows a normal curve with m = 41 minutes, s = 12 minutes, and the shaded area for x between 20 and 50 minutes. The z@scores that correspond to 20 minutes and to 50 minutes are z1 = 20 - 41 12 = -1.75 and z2 = 50 - 41 12 = 0.75. So, the probability that a shopper will be in the store between 20 and 50 minutes is P120 6 x 6 502 = P1-1.75 6 z 6 0.752 = P1z 6 0.752 - P1z 6 -1.752 = 0.7734 - 0.0401 = 0.7333. (b) Interpretation When 200 shoppers enter the store, you would expect about 20010.73332 = 146.66 ≈ 147 shoppers to be in the store between 20 and 50 minutes. 2. (a) The figure at the left shows a normal curve with m = 41 minutes, s = 12 minutes, and the shaded area for x greater than 35. The z@score that corresponds to 35 minutes is z = 35 - 41 12 = -0.5. So, the probability that a shopper will be in the store more than 35 minutes is P1x 7 352 = P1z 7 -0.52 = 1 - P1z 6 -0.52 = 1 - 0.3085 = 0.6915. (b) Interpretation When 200 shoppers enter the store, you would expect about 20010.69152 = 138.3 ≈ 138 shoppers to be in the store more than 35 minutes. TRY IT YOURSELF 2 What is the probability that the shopper in Example 2 will be in the supermarket between 29 and 56 minutes? When 200 shoppers enter the store, how many shoppers would you expect to be in the store between 29 and 56 minutes? Answer: Page A39 EXAMPLE 2 μ = 41 x Time (in minutes) 10 20 30 40 50 60 70 80
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