handout_5_2_fall11(1)

# handout_5_2_fall11(1) - 5.2 Sampling Distributions for...

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5.2 Sampling Distributions for Counts and Proportions (old 5.1) link between Chapter 4 (probability theory) and rest of book “Statistical inference draws conclusions about a population or process on the basis of data. The data are summarized by statistics such as means, proportions, and the slopes of least-squares regression lines. When the data are produced by random sampling or randomized experimentation, a statistic is a random variable that obeys the laws of probability theory. The link between probability and data is formed by the sampling distributions of statistics. A sampling distribution shows how a statistic would vary in repeated data production. That is, a sampling distribution is a probability distribution that answers the question ‘What would happen if we did this many times?’” Count and sample proportion survey 2500 adults, ask if agree or disagree with: “I support Obama.”→ 1250 agreed random variable X is a count of the occurrences of some outcome in a fixed # of observations sample proportion : p ˆ = X / n X = 1500 p ˆ = (1250/2500) = 0.5 The binomial distributions for sample counts The binomial setting - four components 1. There are a fixed number n of observations. 2. The n observations are all independent 3. Each observation falls into one of just two categories, which for convenience we call “success” and “failure.” 4. The probability of success, call it p , is the same for each observation. Binomial distributions The distribution of the count X of successes in the binomial setting is called the binomial distribution with parameters n and p . The parameter n is the number of observations, and p is the probability of a success on any one observation. The possible values of X are the whole numbers from 0 to n . As an abbreviation, we say that X is B( n , p ). Example #1 Toss a coin 10 times and count the number X of heads. The # of heads we observe has B(10, 0.5).

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handout_5_2_fall11(1) - 5.2 Sampling Distributions for...

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