1. Comparing Two Populations Chapter 9 Gonick Anthony Timpson Geog 3000 Feb 21 st, 2010

2. What are we doing? How do we test questions like: – Does taking aspirin reduce the risk of heart attack – Does this pesticide increase our crop yield – Do men and women make the same wage for the same job To test these questions we compare two independent random samples taken from each population

3. Comparing Success rates In the example of Aspirin and heart attack rate – A large sample (20,000 subjects) – Two groups, placebo and aspirin – Compare directly the rates of heart attack in both groups – P-hat 1 = x 1 /n 1 and P-hat 2 = x 2 /n 2 – Where x = the number of successes in each trial n = the total sample size P-hat = rate of heart attack in sample group

4. Comparing Success rates AttackNo AttackNAttack rate Placebo2391079511034.0217 Aspirin1391089811037.0126 -The observed difference between the two is.0217-.0126=.0091 -The observed difference in risk is.0217/.0126 = 1.72, meaning those who do not take aspirin are 1.72 times more likely to have a heart attack

5. Sampling Distribution for P-hat 1 - P-hat 2 For large samples P-hat 1 - P-hat 2 is ~ normally distributed Remember these equations we will use the results to calculate confidence intervals on the next page – Z= ((P-hat 1 - P-hat 2 ) – (p 1 -p 2 ))/σ (P-hat 1 - P-hat 2 ) – σ(P-hat 1 - P-hat 2 ) = Sqrt(σ 2 (P-hat 1 ) + σ 2 (P-hat 2 ))

6. Confidence Intervals for p 1 -p 2 Standard Error – SE(P-hat 1 - P-hat 2 )= Sqrt(((P-hat 1 )(1- P-hat 1 )/n 1 ) + (P- hat 2 )(1- P-hat 2 )/n 2 )) – For the aspirin test the value SE =.0175 Confidence Interval – The range of values we can expect to see with an expected confidence level, (95% for this test) – use the difference between placebo and aspirin.0091,the z value 1.96 (95% confidence) and the SE.0175 to calculate the expected range in heart attack reduction – Expected rate of heart attack reduction with daily aspirin intake, = (.0091 +- (1.96)(.0175)) * 100%

7. Hypothesis Testing H o = 0Null Hypothesis, aspirin has no effect H a not = 0 Test hypothesis, aspirin has an effect Where Z = Std dev from H o = P-hat 1 - P-hat 2 / SE(P-hat 1 - P-hat 2 )=.0091/.00175 = 5.20 H o = 0 H a = 5.20 Aspirin definitely has an effect on heart attack rates

8. Summary This is one example of comparisons between two populations There are countless other examples, each work in the same way utilizing the same principles to achieve the same goal The end product is a range of values which can be expected given a selected confidence level and sufficient sample size