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lect17v1_2up_1up

# lect17v1_2up_1up - Stat 104 Quantitative Methods for...

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Stat 104: Quantitative Methods for Economists Class 17: Confidence Intervals- One Sample Proportion 1

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What is a sample proportion? square6 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 X i = 1 if y es square6 The true (but unknown) population proportion of people who like the movie is p so that P(X i =1)= p . square6 We can estimate p by using 2 0 if n o \$ p X n X i i n = = = 1 1
1 1 1 0 3 0 1 \$ p = 4 6

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Another Student Survey Sample Member Student ID Do you smoke regularly? Numerical Coding 1 232923 No 0 2 234932 Yes 1 3 Yes 1 4 No 0 : : : : 49 No 0 50 Yes 1 11 ˆ 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 square6 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 ) ˆ ~ , p p p N p n -

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Example square6 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? square6 i.e.: if p = .4 and n = 200, what is P(.40 p .45) ? ^ 6
Example square6 if p = .4 and n = 200, what is P(.40 p .45) ? ˆ p(1 p) .4(1 .4) σ .03464 n 200 p - - = = = Find : ˆ p σ ^ .40 .40 .45 .40 ˆ 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

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Confidence Intervals for Proportions square6 Using the same logic as before for means, a (1- α) 100% confidence interval is given by \$ ( \$ ) p p ± - 1 square6 We always assume n is big (larger than 30) when estimating proportions . 8 \$ / p z n α 2
The critical value square6 The 95% confidence interval is usually used, but some other favorites are 90% and 99% 9

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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) 0.40 1.96 0.40 1.96(0.05) (0.30398,0.49602) - ± = ± = 10 100 So with 95% confidence, we estimate the proportion of all men in Chicago who prefer the Gillette Sensor to be somewhere between 30 and 50 percent (pretty good market share).
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