Ch 7: Sampling Distributions

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Statistics

Sampling Distributions

AP Statistics

Unit 5: Sampling Distributions

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23 Terms

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x bar (the sample mean)

estimates μ (the population mean)

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p hat (the sample proportion)

estimates p (the population proportion)

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Sx (the sample st. dev.)

estimates σ (the population st. dev.

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statistic

describes characteristics of a sample

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parameter

describes characteristics of a population

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sampling variability

different random samples of the same size from the same population product different values for a statistic

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sampling distribution

distribution of values taken by the statistic in all possible samples of the same size from the same population

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unbiased estimator

mean of the sampling distribution is equal to the value of the parameter being estimated

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variability

described by the spread of its sampling distribution — inversely proportional to sample size

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 sampling distribution of the sample proportion p hat

describes the distribution of values taken by the sample proportion p hat in all possible samples of the same size from the same population. 

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mean of the sampling distribution

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" data-is-equatio="1" data-latex="\mu_{\hat{p}}=p"><msub><mi>μ</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo stretchy="false">^</mo></mover></mrow></mrow></msub><mo>=</mo><mi>p</mi></math>


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standard deviation of the sampling distribution (as long as the 10% condition is satisfied)

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" data-is-equatio="1" data-latex="\sigma_{\hat{p}}=\sqrt{\frac{p\left(1-p\right)}{n}}"><msub><mi>σ</mi><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo stretchy="false">^</mo></mover></mrow></mrow></msub><mo>=</mo><msqrt><mfrac><mrow><mi>p</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mn>1</mn><mo>−</mo><mi>p</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow></mrow><mi>n</mi></mfrac></msqrt></math>

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sampling distribution is approximately Normal if

the Large Counts condition is satisfied

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mean of the sampling distribution of p1-p2

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" data-is-equatio="1" data-latex="\mu_{\hat{p}_1-\hat{p}_2\ =\ p_1-p_2}"><msub><mi>μ</mi><mrow data-mjx-texclass="ORD"><msub><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo stretchy="false">^</mo></mover></mrow><mn>1</mn></msub><mo>−</mo><msub><mrow data-mjx-texclass="ORD"><mover><mi>p</mi><mo stretchy="false">^</mo></mover></mrow><mn>2</mn></msub><mtext></mtext><mo>=</mo><mtext></mtext><msub><mi>p</mi><mn>1</mn></msub><mo>−</mo><msub><mi>p</mi><mn>2</mn></msub></mrow></msub></math>

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standard deviation of p1-p2 if the 10% condition is met

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shape of the sampling distribution of p1-p2 is approximately Normal if

the Large Counts condition is met for BOTH samples

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sampling distribution of the sample mean x-bar

distribution of values taken by the sample mean x-bar in all possible samples of the same size from the same population

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mean of the sampling distribution of x-bar

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" data-is-equatio="1" data-latex="\mu_{\overline{x}}=\mu"><msub><mi>μ</mi><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo accent="true">―</mo></mover></mrow></msub><mo>=</mo><mi>μ</mi></math>

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standard deviation of the sampling distribution of x-bar as long as the 10% condition is met

<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" data-is-equatio="1" data-latex="\sigma_{\overline{x}}=\frac{\sigma}{\sqrt{n}}"><msub><mi>σ</mi><mrow data-mjx-texclass="ORD"><mover><mi>x</mi><mo accent="true">―</mo></mover></mrow></msub><mo>=</mo><mfrac><mi>σ</mi><msqrt><mi>n</mi></msqrt></mfrac></math>

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central limit theorem

when n is sufficiently large (> or =30), the sampling distribution of the sample mean xis approximately Normal.

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mean of the sampling distribution of x1-x2

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standard deviation of the sampling distribution of x1-x2 as long as the 20% condition is met

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the shape of the sampling distribution of x1-x2 is Normal if

both populations are normal OR if the CLT is satisfied