Margin of Error for Proportions

Transcription

1 for Proportions Gene Quinn for Proportions p.1/8

2 An interval estimate for a population proportion p is often reported not as a confidence interval, but as a margin of error. for Proportions p.2/8

3 An interval estimate for a population proportion p is often reported not as a confidence interval, but as a margin of error. Definition: The margin of error for an estimate ˆp of a population proportion is defined to be 1/2 the width of the widest possible 95% confidence interval for a sample size of n. The margin of error is usually quoted as a percentage. for Proportions p.2/8

4 Suppose a series of n independent Bernoulli trials with unknown probability of success p produces k successes. As we have seen, an approximate 95% confidence interval for p is given by: k n 1.96 k n (1 k n ) n, k n k n (1 k n ) n for Proportions p.3/8

5 If [L,U] is the 95% confidence interval for p, half the width of the interval is: U L = k k 2 n n (1 n k) k k n n 1.96 n (1 n k) n for Proportions p.4/8

6 If [L,U] is the 95% confidence interval for p, half the width of the interval is: U L = k k 2 n n (1 n k) k k n n 1.96 n (1 n k) n Collecting terms and writing ˆp for k/n, this becomes U L 2 = 3.92 ˆp(1 ˆp) n for Proportions p.4/8

7 It is easy to verify that for 0 ˆp 1, ˆp(1 ˆp) is maximized when ˆp = 1 2 for Proportions p.5/8

8 It is easy to verify that for 0 ˆp 1, ˆp(1 ˆp) is maximized when ˆp = 1 2 Taking the derivative with respect to ˆp gives d dˆp (ˆp(1 ˆp)) = d dˆp (ˆp ˆp 2 ) = 1 2ˆp for Proportions p.5/8

9 It is easy to verify that for 0 ˆp 1, ˆp(1 ˆp) is maximized when ˆp = 1 2 Taking the derivative with respect to ˆp gives d dˆp (ˆp(1 ˆp)) = d dˆp (ˆp ˆp 2 ) = 1 2ˆp It is clear that the derivative is zero when ˆp = 1/2 and that this represents a maximum because the second derivative is 2 < 0. for Proportions p.5/8

10 We can obtain the widest possible 95% confidence interval for p by substituting 1/4 for ˆp(1 ˆp): ( max U L ) ˆp(1 ˆp) 1 = max 3.92 = n 4n for Proportions p.6/8

11 We can obtain the widest possible 95% confidence interval for p by substituting 1/4 for ˆp(1 ˆp): ( max U L ) ˆp(1 ˆp) 1 = max 3.92 = n 4n The margin of error is half the width of the widest possible 95% confidence interval, or ( ) = n 2 n for Proportions p.6/8

12 Example: A poll of 1, 000 houseolds finds 47 with an unemployed adult family member. What is the margin of error for the poll? for Proportions p.7/8

13 Example: A poll of 1, 000 houseolds finds 47 with an unemployed adult family member. What is the margin of error for the poll? Actually the only information we need is that n = Then margin of error = = The margin of error is 3.1%. for Proportions p.7/8

14 Example: A survey of 450 college students is taken to determine the proportion that have taken a statistics course. What is the margin of error for the poll? for Proportions p.8/8

15 Example: A survey of 450 college students is taken to determine the proportion that have taken a statistics course. What is the margin of error for the poll? Using the fact that n = 450, margin of error = =.0462 The margin of error is 4.6%. for Proportions p.8/8