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Lecture 17

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Jeff Racine

Lecture 17 Confidence Intervals σ 2 The T-Distribution and Confidence Intervals when is Unknown So far we have examined the sampling distribution of the sample mean. We have the result that ́ σ 2 X N μ( ) n If 1. The population is normally distributed, or 2. The sample size is large, say n ≥30 Under 1. X hasanexactly normal distribution If 2. Holds, thenX has an approximately normal distribution 2 Confidence Intervals for μwhenσ isUnknown Usually the variance is unknown, and we must estimate it using2 s forσ causesaminor problem, This substitution of because the z-score no longer has a normal distribution 2 2 If we replace σ bys , the ratio ́ t= X−μ s2 √ n Is called the t-score and it has a t-distribution with n-1 degrees of freedom ́ s2 A (1-a) x 100% confidence interval foristhereforegivenby X±t a√ n 2 Note that the t-score follows a t-distribution when 1. The population is normal and the sample size is large or small 2. The population is non-normal and sample is large (i.e., ≥30−thisresultisonlyanapproximation¿ ta X 1−X 2± √ variance1+variance2 2 Hypothesis Testing A hypothesis is a statement about the value or set of value or set of values that a parameter or group of parameters can assume (proposition tentatively as possibly true) Often we might, for example, like to test a hypothesis about an unknown population
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