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IPS6eCh05_1bb

# IPS6eCh05_1bb - Sampling Distributions For Counts and...

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Sampling Distributions For Counts and Proportions IPS Chapter 5.1 © 2009 W. H. Freeman and Company

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Objectives (IPS Chapter 5.1) Sampling distributions for counts and proportions Binomial distributions for sample counts Binomial distributions in statistical sampling Binomial mean and standard deviation Sample proportions Normal approximation Binomial formulas
Reminder: the two types of data Quantitative Something that can be counted or measured and then averaged across individuals in the population (e.g., your height, your age, your IQ score) Categorical Something that falls into one of several categories. What can be counted is the proportion of individuals in each category (e.g., your gender, your hair color, your blood type—A, B, AB, O). How do you figure it out? Ask: What are the n individuals/units in the sample (of size “ n ”)? What is being recorded about those n individuals/units? Is that a number ( quantitative) or a statement ( categorical)?

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Binomial distributions for sample  counts Binomial distributions are models for some categorical variables, typically 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 statistically 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.
We express a binomial distribution for the count X of successes among n observations as a function of the parameters n and p: B ( n,p ). The parameter n is the total number of observations. The parameter p is the probability of success on each observation. The count of successes X can be any whole number between 0 and n . A coin is flipped 10 times. Each outcome is either a head or a tail. The variable X is the number of heads among those 10 flips, our count of “successes.” On each flip, the probability of success, “head,” is 0.5. The number X of heads among 10 flips has the binomial distribution B ( n = 10, p = 0.5).

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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, not its magnitude. 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. Is a person ambitious or not? The binomial distribution describes the number of ambitious persons, not how ambitious they are.
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IPS6eCh05_1bb - Sampling Distributions For Counts and...

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