The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STOR 155 Introductory Statistics Lecture 18 Inference for a single proportion Section 8 1 4 5 11 Lecture 18 1 Normal Approximation for Counts and Proportions 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 4 5 11 Lecture 18 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 4 5 11 Lecture 18 z 3 Hypothesis Testing for p For a hypothesized value p0 we want to test H0 p p0 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 p0 0 q where 0 4 5 11 p 0 1 p 0 n Lecture 18 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 p0 P Z z for 1 sided lower tail Ha p p0 2 P Z z for 2 sided Ha p6 p0 Step 4 Compare the P value with the significance level and draw your conclusion 4 5 11 Lecture 18 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 4 5 11 Lecture 18 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 4 5 11 Lecture 18 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 4 5 11 Lecture 18 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 np0 10 4 5 11 and n 1 p0 10 Lecture 18 9
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