Example H \u03b2 c where c is a value of interest in thecontext of a regression

Example h β c where c is a value of interest in

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Example: H 0 : β = c , where c is a value of interest in the context of a regression model. A common value for c is 0
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Components 2 : Alternative Hypothesis 2 Alternative Hypothesis H 1 : Logical alternative hypothesis paired with the null H 0 (that we will “accept” if the null hypothesis is rejected) Three possible types of H 1 for H 0 : β = c : 1 H 1 : β 6 = c 2 H 1 : β > c 3 H 1 : β < c Testing H 0 : β c versus H 0 : β < c is equivalent to testing H 0 : β = c versus H 0 : β < c . Testing H 0 : β c versus H 0 : β > c is equivalent to testing H 0 : β = c versus H 0 : β > c .
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Summary Interval Estimation Hypothesis Testing Examples of Hypothesis Tests Key Words Components 3 : The Test Statistic 3 Test statistic: statistic (realized random variable) whose probability distribution is completely known when the null hypothesis is true ! We know that t = b - β b se ( b ) t ( N - 2) has a t-distribution based on our model assumptions. Thus, if H 0 : β = c is true , it follows that: t = b - c b se ( b ) t ( N - 2) If the null hypothesis is not true , then the t -statistic above does not have a t -distribution. Rodrigo Pinto Hypothesis Testing
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Component 4: Critical Value and Rejection Region 4 Critical Value and Rejection Region: The alternative hypothesis (and the significance level α ) are used to define a critical value t c This critical value defines a Rejection Region that depends on the Alternative Hypothesis H 1 : β k 6 = c implies a double -sided region (two-tail) H 1 : β k > c implies a right -sided region (right-tail) H 1 : β k < c implies a left -sided region (left-tail)
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: :
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One-tail Test for H 0 : β = c versus H 1 : β < c 1 To test the Null H 0 : β k = c , against Alternative H 1 : β k < c , 2 Rejection Rule: ˆ t = ˆ b - c b se ( b ) | {z } Test Statistic < t c ( α, N - 2) | {z } Critical Value t ( m ) Reject H 0 : β k c Do not reject H 0 : β k c 0 t c t ( α , N 2) α 3.3 The rejection region for a one-tail test of H : b ¼ c against H : b < c .
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Two-tail Test for H 0 : β = c against H 1 : β 6 c c c c 2)
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Interpreting the Rejection Region and Significance Level The rejection region consists of values that are unlikely and that have low probability of occurring when the null hypothesis is true The chain of logic is: If a value of the test statistic is obtained that falls in a region of low probability, then it is unlikely that the test statistic has the assumed distribution, and thus it is unlikely that the null hypothesis is true If the alternative hypothesis is true, then values of the test statistic will tend to be unusually large (or unusually small)
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  • Winter '07
  • SandraBlack
  • Statistical hypothesis testing

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