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# Section5.1posted - STOR 155 Section 2 Tuesday Section 5.1...

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STOR 155, Section 2 Tuesday, March 30, 2010 Section 5.1

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Chapter 5: Sampling distributions Introduction: Sampling distributions 5.1: Sampling distributions for counts and proportions 5.2: The sampling distribution of a sample mean
Introduction: Sampling distributions Example: the sampling distribution of a sample mean Heights in inches of women aged 18 – 24 are N (64.5,2.5) So, the height of a single woman chosen at random from this population is a random variable with the N (64.5,2.5) density curve. The histogram of many such random choices will have the same shape as the N (64.5,2.5) density curve. But if we choose a random sample of n = 5 women from this population, their mean height is a random variable with a different distribution. (Will see later that it’s N (64.5,1.12).) This distribution is the sampling distribution of the sample mean. The histogram of many such sample means will have the shape of the density curve of the sampling distribution.

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60 65 70 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 density curve of N(62.5,2.5) 60 65 70 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 10,000 single observations from a N(62.5,2.5) population 60 65 70 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 density curve of the sampling distribution of the mean of a smple of size 5 60 65 70 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 means of 10,000 samples of size 5 from a N(62.5,2.5) population
Introduction: Sampling distributions Example: the sampling distribution of a count 49% of babies born in the US are females. So a bar graph of the sexes of many randomly-chosen babies will have two bars: one of height .49 and one of height .51. But if we choose a random sample of 10 babies and let X be the number of females, then X is a random variable whose possible values are 0, 1, 2, 3, …, 9, 10, with certain probabilities. This is the sampling distribution of the count of female babies in a sample.

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0 1 2 3 4 5 6 7 8 9 10 0 500 1000 1500 2000 2500 numbers of females in 10,000 samples of size 10 Introduction: Sampling distributions Example: the sampling distribution of a count
5.1 Sampling distributions for counts and proportions A random variable X has a binomial probability distribution if X arises in the following setting: We make n independent observations. Each observation falls into one or the other of two categories. For convenience we label one category “success” and the other “failure”. The probability of a success is the same number, denoted p , for each observation. X is the number of successes in the n observations.

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Toss a fair coin 10 times and count the number of heads.
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Section5.1posted - STOR 155 Section 2 Tuesday Section 5.1...

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