1. 1 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman M ARIO F. T RIOLA E IGHTH E DITION E LEMENTARY S TATISTICS Section 6-5 Estimating a Population Proportion

2. 2 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 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. ˆˆ

3. 3 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman q = 1 - p = sample proportion of x failures in a sample size of n p =p = ˆ xnxn sample proportion ˆ p =p = population proportion (pronounced ‘p-hat’) of x successes in a sample of size n Notation for Proportions ˆ

4. 4 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Definition Point Estimate The sample proportion p is the best point estimate of the population proportion p. (In other books population proportion can be noted as  ) ˆ

5. 5 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Margin of Error of the Estimate of p zz  ME = n ˆˆ p q

6. 6 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Confidence Interval for Population Proportion p - ME < < + ME where ˆ p ˆ p zz  ME = n ˆˆ p q

7. 7 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Confidence Interval for Population Proportion p - E < < + E p = p + E ( p - E, p + E) ˆ p ˆ p ˆ ˆˆ

8. 8 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Round-Off Rule for Confidence Interval Estimates of p Round the confidence interval limits to three significant digits.

9. 9 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 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%?

10. 10 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 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

11. 11 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Do people lie about voting? b) Find the 95% confidence interval estimate of the proportion

12. 12 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 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.

13. 13 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman ( ) 2 ˆ p q Determining Sample Size  zz ME = p q n ˆ ˆ (solve for n by algebra)  zz n =n = ˆ ME 2

14. 14 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Sample Size for Estimating Proportion p When an estimate of p is known: ˆ ˆ ( ) 2 p q n =n = ˆ E2E2  zz When no estimate of p is known: ( ) 2 0.25 n =n = E2E2  zz

15. 15 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman ME 2 n =n =  zz ( ) 2 (0.25) Two formulas for proportion sample size zz n =n =  ( ) 2 p q ME 2 ˆ ˆ

16. 16 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 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.

17. 17 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman n = [z  /2 ] 2 p q ˆˆ E 2 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.

18. 18 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman n = [z  /2 ] 2 p q E 2 = [1.645] 2 (0.169)(0.831) 0.04 2 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

19. 19 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman = [1.645] 2 (0.169)(0.831) n = [z  /2 ] 2 p q E 2 = 237.51965 = 238 households 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. 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. 0.04 2 ˆˆ

20. 20 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman 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.

21. 21 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman n = [z  /2 ] 2 (0.25) E 2 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.

22. 22 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman n = [z  /2 ] 2 (0.25) E = (1.645) 2 (0.25) 2 0.04 2 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. = 422.81641 = 423 households

23. 23 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman n = [z  /2 ] 2 (0.25) E = (1.645) 2 (0.25) 2 = 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%. 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. 0.04 2

24. 24 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Finding the Point Estimate and E from a Confidence Interval Point estimate of p : p = (upper confidence interval limit) + (lower confidence interval limit) 2 Margin of Error: E = (upper confidence interval limit) - (lower confidence interval limit) 2 ˆ ˆ

25. 25 Chapter 6. Section 6-5. Triola, Elementary Statistics, Eighth Edition. Copyright 2001. Addison Wesley Longman Finding the Point Estimate and E from a Confidence Interval Find the point estimate of p : p =.678 +.214 =.446 2 ˆ ˆ Margin of Error: E =.678 -.214 =.232 2 Given the confidence interval.214< p <.678