Report

ELEMENTARY STATISTICS Section 6-5 Estimating a Population Proportion EIGHTH Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman EDITION MARIO F. TRIOLA 1 Assumptions 1. The sample is a simple random sample. 2. The conditions for the binomial distribution are satisfied 3. The normal distribution can be used to approximate the distribution of sample proportions because np 5 and nq 5 are both satisfied. ˆ ˆ Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 2 Notation for Proportions p= ˆp = nx population proportion sample proportion of x successes in a sample of size n (pronounced ‘p-hat’) qˆ = 1 - pˆ = sample proportion of x failures in a sample size of n Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 3 Definition Point Estimate ˆ The sample proportion p is the best point estimate of the population proportion p. (In other books population proportion can be noted as ) Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 4 Margin of Error of the Estimate of p ME = z pˆ qˆ n Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 5 Confidence Interval for Population Proportion pˆ - ME < p < pˆ + ME where ME = z pˆ qˆ n Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 6 Confidence Interval for Population Proportion pˆ - E < p < pˆ + E p = pˆ + E (pˆ - E, pˆ + E) Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 7 Round-Off Rule for Confidence Interval Estimates of p Round the confidence interval limits to three significant digits. Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 8 Do people lie about voting? • In a survey of 1002 people, 701 people said they voted in a recent election. Voting records showed that 61% of eligible voters actually did vote. Using these results, find the following about the people who “said” they voted: • a) Find the point estimate • b) Find the 95% confidence interval estimate of the proportion • c) Are the survey results are consistent with the actual voter turnout of 61%? Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 9 Do people lie about voting? • In a survey of 1002 people, 701 people said they voted in a recent election. Voting records showed that 61% of eligible voters actually did vote. Using these results, find the following about the people who said they voted: • a) Find the point estimate pˆ x n 701 0.6996 0.700 1002 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 10 Do people lie about voting? • b) Find the 95% confidence interval estimate of the proportion E z / 2 ˆˆ pq 1.96 (.70)(.30) n .0284 1002 pˆ E p pˆ E .7 .0284 p .7 .0284 .671 p .728 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 11 Do people lie about voting? • c) Are the survey results consistent with the actual voter turnout of 61%? We are 95% confident that the true proportion of the people who said they vote is in the interval 67.1% <p<72.8% . Because 61% does not fall inside the interval, we can conclude our survey results are not consistent with the actual voter turnout. Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 12 Determining Sample Size ME = z pˆ qˆ n (solve for n by algebra) n= ( z pˆ qˆ 2 ) ME2 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 13 Sample Size for Estimating Proportion p ˆ When an estimate of p is known: n= ( 2 pq ) z ˆ ˆ E2 When no estimate of p is known: n= ( 2 0.25 ) z E2 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 14 Two formulas for proportion sample size n= n= ( z )2 pˆ qˆ ME2 ( z (0.25) 2 ) ME2 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 15 Example: We want to determine, with a margin of error of four percentage points, the current percentage of U.S. households using e-mail. Assuming that we want 90% confidence in our results, how many households must we survey? A 1997 study indicates 16.9% of U.S. households used e-mail. Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 16 Example: We want to determine, with a margin of error of four percentage points, the current percentage of U.S. households using e-mail. Assuming that we want 90% confidence in our results, how many households must we survey? A 1997 study indicates 16.9% of U.S. households used e-mail. ˆˆ n = [z/2 ]2 p q E2 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 17 Example: We want to determine, with a margin of error of four percentage points, the current percentage of U.S. households using e-mail. Assuming that we want 90% confidence in our results, how many households must we survey? A 1997 study indicates 16.9% of U.S. households used e-mail. ˆˆ n = [z/2 ]2 p q E2 = [1.645]2 (0.169)(0.831) 0.042 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 18 Example: We want to determine, with a margin of error of four percentage points, the current percentage of U.S. households using e-mail. Assuming that we want 90% confidence in our results, how many households must we survey? A 1997 study indicates 16.9% of U.S. households used e-mail. ˆˆ n = [z/2 ]2 p q E2 = [1.645]2 (0.169)(0.831) 0.042 = 237.51965 = 238 households To be 90% confident that our sample percentage is within four percentage points of the true percentage for all households, we should randomly select and survey 238 households. Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 19 Example: We want to determine, with a margin of error of four percentage points, the current percentage of U.S. households using e-mail. Assuming that we want 90% confidence in our results, how many households must we survey? There is no prior information suggesting a possible value for the sample percentage. Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 20 Example: We want to determine, with a margin of error of four percentage points, the current percentage of U.S. households using e-mail. Assuming that we want 90% confidence in our results, how many households must we survey? There is no prior information suggesting a possible value for the sample percentage. n = [z/2 ]2 (0.25) E2 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 21 Example: We want to determine, with a margin of error of four percentage points, the current percentage of U.S. households using e-mail. Assuming that we want 90% confidence in our results, how many households must we survey? There is no prior information suggesting a possible value for the sample percentage. n = [z/2 ]2 (0.25) E2 = (1.645)2 (0.25) 0.042 = 422.81641 = 423 households Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 22 Example: We want to determine, with a margin of error of four percentage points, the current percentage of U.S. households using e-mail. Assuming that we want 90% confidence in our results, how many households must we survey? There is no prior information suggesting a possible value for the sample percentage. n = [z/2 ]2 (0.25) E2 = (1.645)2 (0.25) 0.042 = 422.81641 = 423 households With no prior information, we need a larger sample to achieve the same results with 90% confidence and an error of no more than 4%. Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 23 Finding the Point Estimate and E from a Confidence Interval ˆ (upper confidence interval limit) + (lower confidence interval limit) Point estimate of p: ˆ p= 2 Margin of Error: E = (upper confidence interval limit) - (lower confidence interval limit) 2 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 24 Finding the Point Estimate and E from a Confidence Interval Given the confidence interval .214< p < .678 ˆ Find the point estimate of p: ˆ p = .678 + .214 = .446 2 Margin of Error: E = .678 - .214 = .232 2 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 25