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PSLS.PPT.Ch19

# PSLS.PPT.Ch19 - proportion PSLS chapter 19 2009 W.H Freeman...

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Inference about a population  proportion PSLS chapter 19 © 2009 W.H. Freeman and Company

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Objectives (PSLS chapter 19) Inference for a population proportion Conditions for inference on proportions The sample proportion (p hat ) The sampling distribution of Significance test for a proportion Confidence interval for p Sample size for a desired margin of error p ˆ p ˆ
Conditions for inference on proportions Assumptions: 1. The data used for the estimate are a random sample from the population studied. 2. The population is at least 20 times as large as the sample. This ensures independence of successive trials in the random sampling. 3. The sample size n is large enough that the shape of the sampling distribution is approximately Normal. How large depends on the type of inference conducted.

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We choose 50 people in an undergrad class, and find that 10 are Hispanic: p hat = (10)/(50) = 0.2 (proportion of Hispanics in sample) You treat a group of 120 Herpes patients given a new drug; 30 get better: p hat = (30)/(120) = 0.25 (proportion of patients improving in sample) The sample proportion  p ̂ We now study categorical data and draw inference on the proportion, or percentage, of the population with a specific characteristic. If we call a given categorical characteristic in the population “success,” then the sample proportion of successes, ( p hat ) is: sample in the ns observatio of count sample in the successes of count ˆ = p ˆ p
Sampling distribution of  p ̂ The sampling distribution of p hat is never exactly Normal. But for large enough sample sizes, it can be approximated by a Normal curve.

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Implication for estimating proportions Therefore, we won’t need to use a t-distribution (unlike with inference for means) . The mean and standard deviation (width) of the sampling distribution are both completely determined by p and n .
Significance test for  p When testing: H 0 : p = p 0 (a given value we are testing) z = ˆ p - p 0 p 0 (1 - p 0 ) n If H 0 is true, the sampling distribution is known

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