helpsession_module6_S18.pdf - PHC 4069 Biostatistics in...

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PHC 4069 Biostatistics in Society Module 6 Help Session Sampling distributions for sample statistics Prepared by: Hanze Zhang Presented by: Ying Ma
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Sampling distribution of a sample statistic What is is: the probability distribution for the values of the sample statistic based on a random sample that exhibits random sampling variation among repeated samples taken from the same population What it shows: Range of possible values for the sample statistic Probabilities associated with each of these values or with a range of values Ex: sampling distribution of the mean weight in a population of university students.
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Sample proportion p Example: What proportion (P) of USF students have blond hair? In order to find out: Obtain a random sample of n USF students (Random phenomenon) Determine for each student in the sample if they do or do not have blond hair (Yes or no), in other words... Count the number of students (X) who have blond hair in the sample of n USF students (Binomial random variable) Compute the sample proportion p=X/n ^ ^ ^
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Sample proportion p The sample proportion p is an estimate of P = true population proportion This is directly related to the count X (binomial random variable) p=X/n ^ ^ ^
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Sample proportion p In module 5: the probability distribution for X=number of hispanic students in a sample of n=3 students The sample space for the count X: {0, 1, 2, 3} (discrete) The sample space for p (X/n): {0, 0.33, 0.67, 1.0} As the sample size (n) increases, the values of p look more like a continuous variable, with a range of 0 to 1.0 Ex: with n=100 the sample space for p (X/n): {0.01, 0.02, ... 0.99, 1.0} ^ ^ ^ ^
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From last week’s help session:
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Sample proportion p Lecture 6.1: Sampling Distribution for Proportion of “Heads” Out of 10 Coin Tosses ^ 0 0.05 0.1 0.15 0.2 0.25 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Sample Proportion P [X]
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Sample proportion p From last week’s help session: the normal distribution curve for continuous variables ^
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Normal approximation to a binomial distribution If the sample size is sufficiently large, the sampling distribution of p becomes approximately normal : If n P ≥ 10 and n(1 P ) ≥ 10 Center”: E( p ) = P “Spread”: σ p = 𝑃(1−𝑃) 𝑛 E( p ) is the expected value of the sample proportion, p , and P is the proportion in the underlying population ^ ^ ^
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Normal approximation to a binomial distribution (lecture 6,2) From lecture 6.2: 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0 0.1 0.2 0.3 0.4 0.5 Sample Proportion P [X] Binomial (n=10, P=0.1) Normal (μ=0.1, σ=0.095)
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