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A part of the soci708 course notes at the university of north carolina, chapel hill. It covers the concepts of sampling distributions, the binomial distribution, and the law of large numbers. Examples, simulations, and formulas for calculating binomial probabilities.
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Module 5 Sampling Distributions^1
François Nielsen
University of North Carolina Chapel Hill
Fall 2009
(^1) Adapted in part from slides for the course Quantitative Methods in Sociology
(Sociology 6Z3) taught at McMaster University by Robert Andersen (now at University of Toronto)
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Family Were Refugees from Antwerp; Ars Conjectandi published 1713)
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Note implausible sequence of proportions (see later)
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… (^) Jacob Bernouilli contributed to the discovery of a major phenomenon of probability, the Law of Large Numbers: … (^) In the long run, the proportion of a certain outcome of a random trial (say, head turns up when tossing a coin) will tend to stabilize to a stable value … (^) But outcome of one trial is independent of previous outcomes … (^) This is counterintuitive: … (^) People naturally tend to believe in a sort of Law of Small Numbers … (^) People do not normally expect the long runs of the same outcome (say, heads in tossing a coin) that occur in true random processes
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Population & Sample
… (^) The population distribution of a variable is:
… (^) A statistic (e.g., x, ˆp, b 1 ) calculated from a random sample or randomized experimental group is a random variable … (^) The probability distribution of a statistic is its sampling distribution … (^) In remainder of Module 6 we look at the sampling distributions of: … (^) counts & proportions … (^) sample means
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Count X & Proportion ˆp
… (^) In general X is a count of the occurrence of some outcome in a xed number of observations n … (^) E.g., in an agricultural experiment n plants are treated for a fungus; the number X of plants with the fungus is a random variable … (^) The sample proportion is ˆp = X / n … (^) E.g., in the experiment X = 9 out of n = 32 plants have the fungus. The sample proportion is
ˆp =
9 32
… (^) The binomial setting is:
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Binomial Distribution
… (^) Choosing an SRS (without replacement) from a population with proportion p of successes is not exactly a binomial setting … (^) E.g., draw 10 cards from a deck, with red card a success. Then probability of red on second card is not independent of color of rst card … (^) However, if the population is much larger than the sample say, 20 times as large the count X of successes in an SRS of size n has approximately the binomial distribution B(n, p) … (^) E.g., draw sample with n = 200 from about 8,000 graduate students at UNC. Success is: student is female. Suppose p = 0.57. Then number of females X is distributed (almost exactly) as B(200, 0.57)
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Finding Binomial Probabilities (1)
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Finding Binomial Probabilities (3)
n! k!(n − k)! for k = 0, 1,... , n. In the formula the factorial n! for any positive integer is de ned as
n! = n × (n − 1 ) × (n − 2 ) ×... × 2 × 1
and also 0! = 1. … (^) Binomial Probability If X has distribution B(n, p), the binomial probability that X = k (for k = 0, 1,... , n) is
P(X = k) =
n k
pk( 1 − p)n−k
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Origin of the Binomial Formula
… (^) Origin of the binomial formula
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Mean & Standard Deviation of Count & Proportion
… (^) If a count X has the binomial distribution B(n, p), then
μ X = np
σ X =
p np( 1 − p)
… (^) Our estimator of the proportion p of successes in the population is the sample proportion
ˆp =
count of successes in sample size of sample
n … (^) If ˆp is the sample proportion of successes in an SRS of size n from a large population with proportion p of successes^2
μ ˆp = p
σ ˆp =
r p( 1 − p) n (^2) Check this follows from the rules for linear functions of random variables.
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Normal Approximation of Counts & Proportions
… (^) Implications of mean and standard deviation of ˆp
q p( 1 −p) n implies that to divide the standard deviation of^ ˆp by half one must multiply n by four … (^) Normal approximation for counts & proportions: … (^) In an SRS of size n from a large population, when n is large
X is approximately N
np,
p np( 1 − p)
ˆp is approximately N p,
r p( 1 − p) n
where p is the proportion of successes in the population, and X and ˆp = X / n are the count & proportion of successes in the sample, respectively
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Normal Approximation of Counts & Proportions
… (^) E.g., SRS of n = 200 from population of 8,000 UNC graduate students with proportion females p = .57. What is P(ˆp ≤ 0.5) (i.e., the sample has fewer females than males)? … (^) np = 200 × 0.57 = 114 > 10 and n( 1 − p) = 200 × 0.43 = 86 > 10 so rule of thumb is satis ed … (^) Using binomial probabilities: X is distributed as B(200, 0.57). ˆp = 0.5 correspond to X = 100. P(X ≤ 100 ) = 0.02734091 or .027. … (^) Using the normal approximation: μ ˆp = p = 0.57;
σ ˆp =
q p( 1 −p) n =^ 0.03500714;
P(ˆp ≤ 0.5) = P
ˆp − 0.
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Experimental Study of the Sampling Distribution of x with n = 3, n = 10, n = 100
… (^) (a) Population distribution of X (income) … (^) Distribution of x for 600 samples: … (^) (b) n = 3 … (^) (c) n = 10 … (^) (d) n = 100
