# 5.1 - Samplingdistributions forcountsandproportions(5.1)...

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Sampling distributions for counts and proportions  (5.1)

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Sampling distributions for counts and proportions Sampling distribution of a count Sampling distribution of a proportion Binomial distributions Normal approximation
Binomial distribution A coin with P (H) = 0.3 is tossed independently 5 times. Let X be the number of heads observed. Then X is a discrete random variable with possible values 0, 1, 2, 3, 4, 5, P (X = 0) = P (TTTTT) = (0.7) 5 P (X = 1) = P (HTTTT) + P (THTTT) + P (TTHTT) + P (TTTHT) + P (TTTTH) = 5(0.3)(0.7) 4 P (X = 2) = P (HHTTT) + P (HTHTT) + P (HTTHT) + P (HTTTH) + …. . + P (TTTHH) = 10(0.3) 2 (0.7) 3

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A coin with P (H) = p is tossed independently n times. Let X be the number of heads observed. Then X is a discrete random variable with possible values 0, 1, … , n. For each k = 0, 1, … , n, P (X = k) = (# of outcomes with k heads and n-k tails) multiplied by p k (1-p) n-k
Binomial coefficient where The binomial coefficient counts the number of ways in which k heads can be arranged among n observations. It’s also the number of ways you can choose k out of n objects.

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Binomial distribution with parameters n and p: B(n,p)
Binomial distributions Binomial distributions are models for random variables representing the number of successes in a series of n trials. The observations must meet these requirements: The total number of observations n is fixed in advance. Each observation falls into just 1 of 2 categories: success and failure. The outcomes of all n observations are independent. All n observations have the same probability of “success,” p . We record the next 50 births at a local hospital. Each newborn is either a boy or a girl; each baby is either born on a Sunday or not.

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Applications for binomial distributions Binomial distributions describe the possible number of times that a particular event will occur in a sequence of observations. They are used when we want to know about the occurrence of an event. In a clinical trial, a patient’s condition may improve or not. We study the number of patients who improved, not how much better they feel. In quality control we assess the number of defective items in a lot of goods.
Software commands: Excel : =BINOMDIST (number_ s , trials, probability_ s, cumulative) Number_ s: number of successes in trials. Trials: number of independent trials. Probability_ s: probability of success on each trial. Cumulative: a logical value that determines the form of the function. TRUE, or 1, for the cumulative P ( X ≤ Number_ s ) FALSE, or 0, for the probability function P ( X = Number_ s ).

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A box contains 10 black balls and 15 white balls. Keep drawing balls randomly from the box with replacement . Suppose 100 random draws are made. Let X be the # of times a black ball is drawn.
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## This note was uploaded on 09/04/2010 for the course STAT 131 taught by Professor Isber during the Spring '08 term at University of California, Berkeley.

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5.1 - Samplingdistributions forcountsandproportions(5.1)...

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