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Unformatted text preview: Stat 104: Quantitative Methods for Economists Class 17: Confidence Intervals One Sample Proportion 1 What is a sample proportion? s Say we survey n people and ask them if they like the movie The Social Network (yes or no). We obtain n responses of the form 1 if y e s s The true (but unknown) population proportion of people who like the movie is p so that P(X i =1)= p . s We can estimate p by using 2 X i = 0 if n o $ p X n X i i n = = = 1 1 1 1 1 3 1 $ p = 4 6 Another Student Survey Sample Member Student ID Do you smoke regularly? Numerical Coding 1 232923 No 2 234932 Yes 1 3 Yes 1 4 No : : : : 49 No 50 Yes 1 11 22%. 50 p = = Suppose that 11 of the 50 students surveyed report that they regularly smoke. The sample proportion is 4 The Central Limit Theorem Works for Proportions s If a random sample of size n is obtained from some population where the probability of having some characteristic is p , then (for large sample sizes) 5 (1 ) ~ , p p p N p n Example s If the true proportion of voters who support Proposition A is p = .4, what is the probability that a sample of size 200 yields a sample proportion between .40 and .45? s i.e.: if p = .4 and n = 200, what is P(.40 p .45) ? ^ 6 Example s if p = .4 and n = 200, what is P(.40 p .45) ? p(1 p) .4(1 .4) .03464 n 200 p = = = Find : p ^ .40 .40 .45 .40 P(.40 .45) P z .03464 .03464 P(0 z 1.44) p = = Convert to standard normal: Use standard normal table: P(0 z 1.44) = .4251 7 Confidence Intervals for Proportions s Using the same logic as before for means, a (1 ) 100% confidence interval is given by s We always assume n is big (larger than 30) when estimating proportions . 8 $ $ ( $ ) / p z p p n  2 1 The critical value s The 95% confidence interval is usually used, but some other favorites are 90% and 99% 9 Example: A marketing research firm contacts a random sample of 100 men in Chicago and finds that 40% of them prefer the Gillette Sensor razor to all other brands. The 95% C.I. for the proportion of all men in Chicago who prefer the Gillette Sensor is determined as follows: 0.40(1 0.40) .40 1.96 0.40 1.96(0.05) (0.30398,0.49602) = = 10 0.40 1.961....
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This note was uploaded on 03/27/2012 for the course STATS 104 taught by Professor Michaelparzen during the Fall '11 term at Harvard.
 Fall '11
 MichaelParzen

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