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Chapter 18 Sampling distribution models math2200

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Sample proportion Kerry vs. Bush in 2004 –A Gallup Poll 49% for Kerry 1016 respondents –A Rasmussen Poll 45.9% for Kerry 1000 respondents –Why the answers are different?

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Model Let Y be the number of people favoring Kerry in a sample of size n=1000 Y ~ Binomial(n,p) –p: the proportion of people for Kerry in the entire population When n is large, Y can be approximated by Normal model with mean np and variance npq.

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Modeling sample proportion The sample proportion –Normal model with mean p and variance

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Kerry vs. Bush (cont’) –Assume the true population proportion voting for Kerry is 49%. –The sample proportion = Y/n has a normal model with mean 0.49 and standard deviation 0.0158 (n=1000) –Then we know that both 49% and 45.9 % are reasonable to appear (0.459 - 0.49)/0.0158= - 1.962

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Sampling Distribution Model Consider the sample proportion as a random variable instead of a number. The distribution of the sample proportion is called the sampling distribution model for the proportion.

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Left-Handed: Example 13% of the population is left-handed. A 200-seat school auditorium was built with 15 “leftie seats”. In a class of n=90 students, what’s the probability that there will NOT be enough seats for the left-handed students? Let Y be the number of left-handed students in the class. We want to find P(Y>15) = P(Y/n>0.167) = P( >0.167)

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Left-Handed (cont’) Check the conditions –n is large enough –randomization –10% condition The population should have more than 900 students –Success/failure condition np=11.7>10, nq=78.3>10 Approximate by Normal model for Y/n –Mean = 0.13 = p –Standard deviation = = 0.035 P( = Y/n >0.167) =normalcdf(0.167, 1E99, 0.13, 0.035) = 0.1446

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Example: Sampling Distribution of a Mean 10,000 simulations for each graph.

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Central limit theorem (CLT) If the observations are drawn –independently –from the same population (equivalently, distribution) the sampling distribution of the sample mean becomes normal as the sample size increases. The population distribution could be unknown.

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CLT Suppose the population distribution has mean μand standard deviation σ The sample mean has mean μand standard deviation. Let Y1, …, Yn be n independently and identically distributed random variables –E(Y1) = μ –Var(Y1)= σ 2 Then as n increases, the distribution of (Y1+…+Yn)/n tends to a normal model with mean μand standard deviation

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The Fundamental Theorem of Statistics The Central Limit Theorem (CLT) The mean of a random sample has a sampling distribution whose shape can be approximated by a Normal model. The larger the sample, the better the approximation will be.

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Example: Elevator Overloaded Suppose the population distribution of adult weights has mean 175 pounds and sd 25 pounds –the shape is unknown An elevator has a weight limit of 10 persons or 2000 pounds What’s the probability that the 10 people who get on the elevator overload its weight limit?

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Let Xi, i=1,2,…,10 be the weight of the ith person in the elevator Then we want to know P(X1+…+X10>2000) = From the CLT (check the requirement first), we know the distribution of is normal with mean 175 pounds and standard deviation Then

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Standard Error Using the CLT, the distribution of sample proportion is In general, by the CLT the distribution of sample mean of independent sample values is p, and could be unknown in some cases.

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Standard Error If we don’t know or σ, the population parameters, we will use sample statistics to estimate. The estimated standard deviation of a sampling distribution is called a standard error.

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Standard Error (cont.) For a sample proportion, the standard error is For the sample mean, the standard error is

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The Process Going Into the Sampling Distribution Model

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What Can Go Wrong? Don’t confuse the sampling distribution with the distribution of the sample. –When you take a sample, you look at the distribution of the values, usually with a histogram, and you may calculate summary statistics. –The sampling distribution is an imaginary collection of the values that a statistic might have taken for all random samples—the one you got and the ones you didn’t get.

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What Can Go Wrong? (cont.) Beware of observations that are not independent. –The CLT depends crucially on the assumption of independence. –You can’t check this with your data—you have to think about how the data were gathered. Watch out for small samples from skewed populations. –The more skewed the distribution, the larger the sample size we need for the CLT to work.

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Summary Sample proportions or sample means are statistics –They are random because samples vary –Their distribution can be approximated by normal using the CLT Be aware of when the CLT can be used –n is large –If the population distribution is not symmetric, a much larger n is needed The CLT is about the distribution of the sample mean, not the sample itself

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