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AP Statistics Unit 7: Inference for Quantitative Data: Means

Ch 10: Estimating Means with Confidence

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standard error of x

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Statistics

AP Statistics

Unit 7: Inference for Quantitative Data: Means

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standard error of x

$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" data-is-equatio="1" data-latex="SE=\frac{s_x}{\sqrt{n}}"><mi>S</mi><mi>E</mi><mo>=</mo><mfrac><msub><mi>s</mi><mi>x</mi></msub><msqrt><mi>n</mi></msqrt></mfrac></math>$

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Replacing the SD with the SE requires the use of the ____ rather than the normal distribution.

t distribution

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The number of degrees of freedom is equal to

n-1

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A sample meets the condition for being Random if:

the data come from a random sample from the population of interest

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A sample meets the condition for being Independent if:

10% condition is satisfied when sampling without replacement

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A sample meets the condition for being approximately Normal if:

We are told the population is Normally distributed, or
The sample size is large enough, by the Central Limit Theorem (n≥30), or
Graph/sketch the sample data to assess the Normality; look for large skewness or outliers

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equation for one-sample t interval for μ

$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" data-is-equatio="1" data-latex="\overline{x}\pm t^{\ast}\frac{s_x}{\sqrt{n}}"><mover><mi>x</mi><mo accent="true">―</mo></mover><mo>±</mo><msup><mi>t</mi><mrow data-mjx-texclass="ORD"><mo>∗</mo></mrow></msup><mfrac><msub><mi>s</mi><mi>x</mi></msub><msqrt><mi>n</mi></msqrt></mfrac></math>$ (use table B for t* values, and df=n-1)

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Margin of error ____ when sample size increases

decreases

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Margin of error _____ when confidence level increases

increases

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conditions to fulfill for confidence intervals for μ1-μ2

Independent random samples, or from two groups in a randomized experiment
10% condition when sampling without replacement for both n1 and n2
For each sample, follow same rules as above (for 1 sample)

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name of procedure for μ1-μ2

two sample t interval for a difference between two means

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conservative method for calculating degrees of freedom with two samples

take the smaller of n₁-1 and n₂-1

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formula for a two sample t interval for a difference in means

$<math xmlns="http://www.w3.org/1998/Math/MathML" display="block" data-is-equatio="1" data-latex="\left(\overline{x}_1-\overline{x}_2\right)\pm t^{\ast}\sqrt{\frac{s_{_1}^2}{n_1}+\frac{s_{_2}^2}{n_2}}"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><msub><mover><mi>x</mi><mo accent="true">―</mo></mover><mn>1</mn></msub><mo>−</mo><msub><mover><mi>x</mi><mo accent="true">―</mo></mover><mn>2</mn></msub><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>±</mo><msup><mi>t</mi><mrow data-mjx-texclass="ORD"><mo>∗</mo></mrow></msup><msqrt><mfrac><msubsup><mi>s</mi><mrow data-mjx-texclass="ORD"><msub><mi/><mn>1</mn></msub></mrow><mn>2</mn></msubsup><msub><mi>n</mi><mn>1</mn></msub></mfrac><mo>+</mo><mfrac><msubsup><mi>s</mi><mrow data-mjx-texclass="ORD"><msub><mi/><mn>2</mn></msub></mrow><mn>2</mn></msubsup><msub><mi>n</mi><mn>2</mn></msub></mfrac></msqrt></math>$