 # Se y σ y σ y n which is an estimator of the standard

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SE(´Y)=^σ´Y=σYnwhich is an estimator of the standard deviation of the sampling distribution of´Y.2.Compute thet-statistict=´YμY ,0SE(´Y)=´YμY ,0sYn3.Compute thep-value, which is the smallest significance level at which the null hypothesis couldbe rejected, based on the test statistic actually observed; equivalently, thep-value is theprobability of obtaining a statistic, by random sampling variation, at least as different from thenull hypothesis value as is the statistic actually observed, assuming that the null hypothesis iscorrect. In a few words,ifp-value¿α ,reject H0-
Key Concept 5.1: General Form of thet-StatisticIn general, thet-Statistic has the formt=estimatorhypothesized valuestandard error of the estimator=^ββ0SE(^β),(5.1)Testing hypotheses about the slopeβ1Thenulland alternativehypotheses need to bestated precisely before theycanbetested.Theangrytaxpayer'shypothesis is thatβRD=0.
More generally,under the null hypothesis the truepopulation slopeβ1takeson some specific value,β1,0. Under thetwo-sidedalternative,β1does notequalβ1,0.That is,thenullhypothesisandthe two-sided alternative hypothesisareH0:β1=β1,0H1:β1≠ β1,0,(5.2)The first step is to compute the standard error of^β1,SE(^β1). The standard error of^β1is an estimator of^σ^β1, the standard deviation of the sampling distribution of^β1. Specifically,SE(^β1)=σ^β12,(5.3)whereσ^β12=1n×1n2i=1n(Xi´X)2^ui2[1ni=1n(Xi´X)2]2,(5.4)Although the formula forσ^β12is complicated, in applications the standard error iscomputed by regression software so that it is easy to use in practice.The second step is to compute thet-statistic,t=^β1β1,0SE(^β1),(5.5)The third step is to compute thep-value, the probability of observing a value of^β1atleast as different fromβ1,0as the estimate actually computed^β1act, assuming that thenull hypothesis is correct. Stated mathematically,pvalue=PrH0[|^β1β1,0|]>PrH0[|^β1actβ1,0|]¿PrH0[|^β1β1,0SE(^β1)|]>PrH0[|^β1actβ1,0SE(^β1)|]=PrH0(|t|>|tact|),(5.6)wherePrH0denotes the probability computed under the null hypothesis, the secondequality follows by dividing bySE(^β1), andtactis the value of thet-statistic actuallycomputed.

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linear regression model
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