Section 7.1 The Standard Normal Curve 287 The region above a single point has zero width, and thus an area of 0. Therefore, when a population is represented with a probability density curve, the probability of obtaining a prespecified value exactly is equal to 0. For this reason, if X is a continuous random variable, then P(X = a) = 0 for any number a, and P(a < X < b) = P(a ≤ X ≤ b) for any numbers a and b. For any probability density curve, the area under the entire curve is equal to 1, because this area represents the entire population. SUMMARY ∙ A probability density curve represents the probability distribution of a continuous variable. ∙ The area under the entire curve is equal to 1. ∙ The area under the curve between two values a and b has two interpretations: 1. It is the proportion of the population whose values are between a and b. 2. It is the probability that a randomly selected individual will have a value between a and b. EXAMPLE 7.1 Interpret the area under a probability density curve Following is a probability density curve for a population. 0 2 4 6 8 10 12 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0 Area = 0.16 a. What proportion of the population is between 4 and 6? b. If a value is chosen at random from this population, what is the probability that it will be between 4 and 6? c. What proportion of the population is not between 4 and 6? d. If a value is chosen at random from this population, what is the probability that it is not between 4 and 6? Solution a. The proportion of the population between 4 and 6 is equal to the area under the curve between 4 and 6, which is 0.16. b. The probability that a randomly chosen value is between 4 and 6 is equal to the area under the curve between 4 and 6, which is 0.16. c. The area under the entire curve is equal to 1. Therefore, the proportion that is not between 4 and 6 is equal to 1 − 0.16 = 0.84. d. The probability that a randomly chosen value is not between 4 and 6 is equal to the area under the curve that is not between 4 and 6, which is 0.84. Recall: The Rule of Complements says that P (not A) = 1 − P (A). Another way to answer part (d) is to use the Rule of Complements: P(Not between 4 and 6) = 1 − P(Between 4 and 6) = 1 − 0.16 = 0.84
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