# Lec7 - Turn to your neighbor and define the sampling...

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Turn to your neighbor and … - define the sampling distribution of Y - describe how sample size affects this distribution -

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Confidence intervals: Locating an invisible man! The man is within 2 SE’s of the dog about 95% of the time We know where the dog is, and we’d like to estimate where the man is. We know that 95% of the time the man is within 2 SE’s of the dog, so we say that we are 95% confident that the man is in this interval. y y μ μ μ y μ μ 95% confidence interval
Confidence intervals: Mathematically RECALL: If Y has a normal distribution, then Y has a normal distribution. Pr{ 1.96 < Z < 1.96} = .95 (Almost, but not exactly two standard deviations.) Start with the standardized scale: Use the Z table to figure out what we add and subtract from the mean to get 95% of the distribution. y ± 1.96 σ n Therefore, the interval will contain μ for 95% of all samples. RECALL: The standard deviation of the sampling distribution of is Y σ n . Pr{ Y 1.96 σ < μ < Y + 1.96 σ } = .95 So for any normal distribution: Pr{ μ 1.96 σ < Y < μ + 1.96 σ } = .95 μ + 1.96 σ μ 1.96 σ

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BUT, we don’t necessarily know σ If the data come from a normal population, then we can use s to estimate σ , but then we have to change the multiplier from 1.96 to something else in order to estimate the 95% confidence interval. y ± 1.96 σ n Therefore, the interval will contain μ for 95% of all samples. W.S. Gosset (1876-1937), Guinness Brewery, 1908, ‘Student’
BUT, we don’t necessarily know σ If the data come from a normal population, then we can use s to estimate σ , but then we have to change the multiplier from 1.96 to something else in order to estimate the 95% confidence interval.

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