The Central Limit Theorem for Proportions Lecture 26 Sections 8.1 – 8.2 Mon, Mar 3, 2008.

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Presentation transcript:

The Central Limit Theorem for Proportions Lecture 26 Sections 8.1 – 8.2 Mon, Mar 3, 2008

The Central Limit Theorem for Proportions The sampling distribution of p ^ is approximately normal with the following parameters.

The Central Limit Theorem for Proportions The approximation to the normal distribution is excellent if

Example If we gather a sample of 100 males, how likely is it that between 60 and 70 of them, inclusive, wash their hands after using a public restroom? This is the same as asking the likelihood that 0.60  p ^  0.70.

Example Use p = Check that  np = 100(0.66) = 66 > 5,  n(1 – p) = 100(0.34) = 34 > 5. Then p ^ has a normal distribution with

Example So P(0.60  p ^  0.70) = normalcdf(.60,.70,.66,.04737) =

Why Surveys Work Suppose that we are trying to estimate the proportion of the male population who wash their hands after using a public restroom. Suppose the true proportion is 66%. If we survey a random sample of 1000 people, how likely is it that our error will be no greater than 5%?

Why Surveys Work Now we have

Why Surveys Work Now find the probability that p^ is between 0.61 and 0.71: normalcdf(.61,.71,.66,.01498) = It is virtually certain that our estimate will be within 5% of 66%.

Case Study Study confirms aprotinin drug increases cardiac surgery death rate Study confirms aprotinin drug increases cardiac surgery death rate Aprotinin during Coronary-Artery Bypass Grafting and Risk of Death Aprotinin during Coronary-Artery Bypass Grafting and Risk of Death

Case Study In the first study, what is the sample size? What are the variables? Which is explanatory and which is response? What is the relevant sample proportion?

Case Study If the true death rate after 30 days is 2.6%, how likely is a death rate of 6.4% (or higher) in a sample of this size? What are some confounding variables?

Quality Control A company will accept a shipment of components if there is no strong evidence that more than 5% of them are defective. H 0 : 5% of the parts are defective. H 1 : More than 5% of the parts are defective.

Quality Control They will take a random sample of 100 parts and test them. If no more than 10 of them are defective, they will accept the shipment. What is  ? What is  ?