Section 7.4 The Central Limit Theorem for Proportions 321 Objective 2 Use the Central Limit Theorem to compute probabilities for sample proportions Computing Probabilities with the Central Limit Theorem To compute probabilities involving a sample proportion ̂p, use the following procedure: Procedure for Computing Probabilities with the Central Limit Theorem Step 1: Check to see that the conditions np ≥ 10 and n(1 − p) ≥ 10 are both met. If so, it is appropriate to use the normal curve. Step 2: Find the mean ��̂p and standard deviation ��̂p. 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 7.23 Using the Central Limit Theorem to compute a probability Explain It Again Computing probabilities for sample proportions: Computing probabilities for sample proportions with the Central Limit Theorem is the same as computing probabilities for any normally distributed quantity. Use ��p̂ = √ p for the mean and ��p̂ = p(1−p)∕n for the standard deviation. According to a Harris poll taken in September 2013, chocolate is the favorite ice cream flavor for 27% of Americans. If a sample of 100 Americans is taken, what is the probablity that the sample proportion of those who prefer chocolate is greater than 0.30? Solution Step 1: np = (100)(0.27) = 27 ≥ 10, and n(1 − p) = (100)(1 − 0.27) = 73 ≥ 10. We may use the normal curve. 0.27 0.3 z = 0.68 Figure 7.27 Step 2: ��̂p = p = 0.27. ��̂p = √ p(1 − p) n = √ 0.27(1 − 0.27) 100 = 0.044396 Step 3: Figure 7.27 presents the normal curve with the area shaded in. Step 4: We will use Table A.2. We compute the z-score for 0.30. z = ̂p − ��̂p ��̂p = 0.30 − 0.27 0.044396 = 0.68 The table gives the area to the left of z = 0.68 as 0.7517. The area to the right of z = 0.68 is 1 − 0.7517 = 0.2483. The probability that the sample proportion of those who prefer chocolate is greater than 0.30 is 0.2483. EXAMPLE 7.24 Using the Central Limit Theorem to determine whether a given value of p̂ is unusual In the 2012 U.S. presidential election, 51% of voters voted for Barack Obama. If a sample of 75 voters were polled, would it be unusual if less than 40% of them had voted for Barack Obama? Solution We will compute the probability that the sample proportion is less than 0.40. If this probability is less than 0.05, we will say that the event is unusual. Step 1: np = (75)(0.51) = 38.25 ≥ 10, and n(1 − p) = (75)(1 − 0.51) = 36.75 ≥ 10. We may use the normal curve.
navidi_monk_elementary_statistics_2e_ch7-9
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