Chapter20

# The Basic Practice of Statistics (Paper) & Student CD

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Inference about a population proportion BPS chapter 20 © 2006 W.H. Freeman and Company

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Objectives (BPS chapter 20) Inference for a population proportion The sample proportion The sampling distribution of Large sample confidence interval for p Accurate confidence intervals for p Choosing the sample size Significance tests for a proportion p ˆ p ˆ
The two types of data — reminder Quantitative Something that can be counted or measured and then added, subtracted, averaged, etc., across individuals in the population. Example: How tall you are, your age, your blood cholesterol level Categorical Something that falls into one of several categories. What can be counted is the proportion of individuals in each category. Example: Your blood type ( A, B, AB, O ), your hair color, your family health history for genetic diseases, whether you will develop lung cancer How do you figure it out? Ask: What are the n individuals/units in the sample (of size “ n ”)? What’s being recorded about those n individuals/units? Is that a number ( quantitative) or a statement ( categorical)?

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We choose 50 people in an undergrad class, and find that 10 of them are Hispanic: = (10)/(50) = 0.2 (proportion of Hispanics in sample) You treat a group of 120 Herpes patients given a new drug; 30 get better: = (30)/(120) = 0.25 (proportion of patients improving in sample) The sample proportion 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, ,is: sample in the ns observatio of count sample in the successes of count ˆ = p ˆ p ˆ p ˆ p ˆ p
Sampling distribution of The sampling distribution of is never exactly normal. But as the sample size increases, the sampling distribution of becomes approximately normal. p ˆ ˆ p ˆ

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The mean and standard deviation (width) of the sampling distribution are both completely determined by p and n . Thus, we have only one population parameter to estimate, p . Implication for estimating proportions N p , p (1 " p ) n ( ) Therefore, inference for proportions can rely directly on the normal distribution (unlike inference for means, which requires the use of a t distribution with a specific degree of freedom) .
Conditions for inference on p Assumptions: 1. We regard our data as a simple random sample (SRS) from the population. That is, as usual, the most important condition.

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Chapter20 - Inference about a population proportion BPS...

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