Section 7.2 Applications of the Normal Distribution 303 Check Your Understanding 4. A normal population has mean �� = 3 and standard deviation �� = 1. Find the proportion of the population that is less than 1. 0.0228 5. A normal population has mean �� = 40 and standard deviation �� = 10. Find the probability that a randomly sampled value is greater than 53. 0.0968 6. A normal population has mean �� = 7 and standard deviation �� = 5. Find the proportion of the population that is between −2 and 10. 0.6898 Answers are on page 310. Objective 3 Find the value from a normal distribution corresponding to a given proportion Finding the Value from a Normal Distribution Corresponding to a Given Proportion Sometimes we want to find the value from a normal distribution that has a given proportion of the population above or below it. The method for doing this is the reverse of the method for finding a proportion for a given value. In particular, we need to find the value from the distribution that has a given z-score. Explain It Again x = �� + z��: The z-score tells how many standard deviations x is above or below the mean. Therefore, the value of x that corresponds to a given z-score is equal to the mean (��) plus z times the standard deviation (��). Recall that the z-score tells how many standard deviations a value is above or below the mean. The value of x that corresponds to a given z-score is given by x = �� + z�� EXAMPLE 7.16 Finding the value from a normal distribution with a given z-score Heights in a group of men are normally distributed with mean �� = 69 inches and standard deviation �� = 3 inches. a. Find the height whose z-score is 1. Interpret the result. b. Find the height whose z-score is −2.0. Interpret the result. c. Find the height whose z-score is 0.6. Interpret the result. Solution a. We want the height that is equal to the mean plus one standard deviation. Therefore, x = �� + z�� = 69 + (1)(3) = 72. We interpret this by saying that a man 72 inches tall has a height one standard deviation above the mean. b. We want the height that is equal to the mean minus two standard deviations. Therefore, x = �� + z�� = 69 + (−2)(3) = 63. We interpret this by saying that a man 63 inches tall has a height two standard deviations below the mean. c. We want the height that is equal to the mean plus 0.6 standard deviation. Therefore, x = �� + z�� = 69 + (0.6)(3) = 70.8. We interpret this by saying that a man 70.8 inches tall has a height 0.6 standard deviation above the mean. To find the value from a normal distribution that has a given proportion above or below it, we can use either Table A.2 or technology. Following are the steps to find the value that has a given proportion above or below it by using Table A.2. Finding a Normal Value That Has a Given Proportion Above or Below It by Using Table A.2 Step 1: Sketch a normal curve, label the mean, label the value x to be found, and shade in and label the given area. Step 2: If the given area is on the right, subtract it from 1 to get the area on the left. Step 3: Look in the body of Table A.2 to find the area closest to the given area. Find the z-score corresponding to that area. Step 4: Obtain the value from the normal distribution by computing x = �� + z��.
navidi_monk_elementary_statistics_2e_ch7-9
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