1 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Section 7.2 Estimating a Population Proportion Objective Find the confidence interval for a population proportion p Determine the sample size needed to estimate a population proportion p

2 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Definitions The best point estimate for a population proportion p is the sample proportion p Best point estimate : p The margin of error E is the maximum likely difference between the observed value and true value of the population proportion p (with probability is 1–α)

3 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Margin of Error for Proportions E = margin of error p = sample proportion q = 1 – p n = number sample values 1 – α = Confidence Level

4 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Confidence Interval for a Population Proportion p ( p – E, p + E ) ˆ ˆ where

5 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Finding the Point Estimate and E from a Confidence Interval Margin of Error: E = (upper confidence limit) — (lower confidence limit) 2 Point estimate of p : p = (upper confidence limit) + (lower confidence limit) 2

6 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Round-Off Rule for Confidence Interval Estimates of p Round the confidence interval limits for p to three significant digits.

7 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67 Direct Computation Example 1

8 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using StatCrunch Stat → Proportions → One Sample → with Summary Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67

9 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using StatCrunch Enter Values Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67

10 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using StatCrunch Click ‘Next’ Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67

11 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using StatCrunch Select ‘Confidence Interval’ Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67

12 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using StatCrunch Enter Confidence Level, then click ‘Calculate’ Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67

13 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Using StatCrunch From the output, we find the Confidence interval is CI = (0.578, 0.762) Lower Limit Upper Limit Standard Error Example 1 Find the 95%confidence interval for the population proportion If a sample of size 100 has a proportion 0.67

14 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Sample Size Suppose we want to collect sample data in order to estimate some population proportion. The question is how many sample items must be obtained?

15 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Determining Sample Size (solve for n by algebra) ( ) 2 ˆ p q  Z n = ˆ E 2  z E =E = p q ˆ ˆ n

16 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Sample Size for Estimating Proportion p When an estimate of p is known: ˆ ˆ ( ) 2 p q n = ˆ E 2  z When no estimate of p is known: use p = q = 0.5 ( ) n = E 2  z ˆ ˆ ˆ

17 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Round-Off Rule for Determining Sample Size If the computed sample size n is not a whole number, round the value of n up to the next larger whole number. Examples: n = round up to 311 n = round up to 296 n = round up to 114

18 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. A manager for E-Bay wants to determine the current percentage of U.S. adults who now use the Internet. How many adults must be surveyed in order to be 95% confident that the sample percentage is in error by no more than three percentage points when… (a) In 2006, 73% of adults used the Internet. (b) No known possible value of the proportion. Example 2

19 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (a) Given: Given a sample has proportion of 0.73, To be 95% confident that our sample proportion is within three percentage points of the true proportion, we need at least 842 adults. Example 2

20 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. (b) Given: For any sample, To be 95% confident that our sample proportion is within three percentage points of the true proportion, we need at least 1068 adults. Example 2

21 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. Summary Confidence Interval of a Proportion ( p – E, p + E ) E = margin of error p = sample proportion n = number sample values 1 – α = Confidence Level

22 Copyright © 2010, 2007, 2004 Pearson Education, Inc. All Rights Reserved. When an estimate of p is known: ˆ ( ) 2 p q n = ˆ E 2  z When no estimate of p is known (use p = q = 0.5) ( ) n = E 2  z Summary Sample Size for Estimating a Proportion