The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 20 Inference for a single proportion Section 8 1 11 21 06 Lecture 20 1 Normal Approximation for Counts and Proportions Recall this is page 16 in Lecture 15 16 Let X B n p and p X n If n is large then p is approx N p p 1 p n Rule of Thumb np 10 n 1 p 10 11 21 06 Lecture 20 2 Confidence Interval for p Expression p m p m p where the margin of error m z p 1 p n Assumption n is large Confidence level C determines 11 21 06 Lecture 20 z 3 Hypothesis Testing for p For a hypothesized value p 0 we want to test H0 p p 0 versus some alternative 1 sided or 2 sided Recall the 4 steps Step 1 only need to specify Ha Step 2 Test statistic z p p 0 0 q where 0 p0 1 p0 n 11 21 06 Lecture 20 4 Hypothesis Testing for p continued Step 3 The P value will be equal to P Z z for 1 sided upper tail Ha p p 0 P Z z for 1 sided lower tail Ha p p0 2 P Z z for 2 sided Ha p 6 p 0 Step 4 Compare the P value with the significance level and draw your conclusion 11 21 06 Lecture 20 5 Biased one Euro Coin A group of statistics students spun the Belgian one Euro coin 250 times and heads came up 140 times p P H in each spin Claim the coin is biased more specifically p is greater than 0 5 11 21 06 Lecture 20 6 Biased one Euro Coin continued Sample 140 heads among 250 spins of a Belgian one Euro coin a hint p P H in each spin H0 p 0 5 vs Ha p 0 5 one sided upper p z 140 250 0 5 0 5 1 0 5 250 1 897 P value P Z 1 897 1 0 9713 0 0287 Conclude based on a given 11 21 06 Lecture 20 7 Biased one Euro Coin continued What about a 95 CI for p Note Margin of error p p m z p 1 p n 1 96 0 56 1 0 56 250 0 06 95 CI 0 56 0 06 0 56 0 06 0 5 0 62 11 21 06 Lecture 20 8 Take Home Message CI for a single proportion p Margin of error m in CI Hypothesis testing for p 4 steps Assumption large n How large np 0 10 11 21 06 and n 1 p 0 10 Lecture 20 9
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