314 Chapter 7 The Normal Distribution The Central Limit Theorem is the most important result in statistics, and forms the basis for much of the work that statisticians do. Objective 2 Use the Central Limit Theorem to compute probabilities involving sample means Computing Probabilities with the Central Limit Theorem To compute probabilities involving a sample mean ̄x, use the following procedure: Procedure for Computing Probabilities with the Central Limit Theorem Step 1: Be sure the sample size is greater than 30. If so, it is appropriate to use the normal curve. Step 2: Find the mean ��̄ x and standard deviation ��̄ x. Step 3: Sketch a normal curve and shade in the area to be found. Step 4: Find the area using Table A.2 or technology. EXAMPLE 7.20 Using the Central Limit Theorem to compute a probability Based on data from the U.S. Census, the mean age of college students in 2011 was �� = 25 years, with a standard deviation of �� = 9.5 years. A simple random sample of 125 students is drawn. What is the probability that the sample mean age of the students is greater than 26 years? Solution Step 1: The sample size is 125, which is greater than 30. We may use the normal curve. Step 2: We compute ��̄ x and ��̄ x. ��̄ x = �� = 25 ��̄ x = �� √ n = 9.5 √ 125 = 0.85 Step 3: Figure 7.23 presents the normal curve with the area of interest shaded. Step 4: We will use Table A.2. We compute the z-score for 26. 26 z = 1.18 25 Figure 7.23 CAUTION When computing the z-score for the distribution of x̄, be sure to use the standard deviation ��x̄ , rather than ��. z = x − ��̄ x ��̄ x = 26 − 25 0.85 = 1.18 The table gives the area to the left of z = 1.18 as 0.8810. The area to the right of z = 1.18 is 1 − 0.8810 = 0.1190. The probability that the sample mean age of the students is greater than 26 years is 0.1190. EXAMPLE 7.21 Using the Central Limit Theorem to determine whether a given value of x is unusual Hereford cattle are one of the most popular breeds of beef cattle. Based on data from the Hereford Cattle Society, the mean weight of a one-year-old Hereford bull is 1135 pounds, with a standard deviation of 97 pounds. Would it be unusual for the mean weight of 100 head of cattle to be less than 1100 pounds? 1100 z = −3.61 1135 Figure 7.24 Solution We will compute the probability that the sample mean is less than 1100. We will say that this event is unusual if its probability is less than 0.05. Step 1: The sample size is 100, which is greater than 30. We may use the normal curve. Step 2: We compute ��̄ x and ��̄ x. ��̄ x = �� = 1135 ��̄ x = �� √ n = 97 √ 100 = 9.7 Step 3: Figure 7.24 presents the normal curve. We are interested in the area to the left of 1100, which is too small to see.
navidi_monk_elementary_statistics_2e_ch7-9
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