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Unformatted text preview: Probability and Statistics in the Life Sciences (Spring 2010) AMS 110.02 Lecture 11 (chap7) Donghyung Lee Chapter 7 (supplements) Tests About a Population Mean Case I : A Normal population with Known σ /2 /2 : : : / ( ) : ( ) ( ) : ( ) ( ) : a a a Null hypothesis H x Test statistic value z n Alternative Hypothesis Rejection region for level test a H z z upper tailed test b H z z lower tailed test c H either z z or z z α α α α μ μ μ σ α μ μ μ μ μ μ = = ≥ < ≤  ≠ ≥ ≤  ( ) two tailed test Chapter 7 (supplements) Tests About a Population Mean 2 1 2 2 2 2 *** ( ): : : *** ~ ( , ); ... : ~ ( ,( ) ) ( ,( / ) ) ~ (0,1 ) / ** ** : , ~ (0,1 ) / ( a n X X Case I a H H X N is known X X a random sample of size n from the normal population X N N n X Z N n Test Statistic X Under H Z N n obtained by μ μ μ μ μ σ σ μ σ μ σ μ σ μ μ μ σ = = = = = = ) standardizing X under the assumption that H is true Chapter 7 (supplements) Tests About a Population Mean , . ( ), . , a a If the distance between sample mean x and is too great in a direction consistent with H the null hypothesis should be rejected For Case I a a x value less than certainly does not provide support for H Similarly if a x value that exc μ μ ( ) . , a eeds by only a small amount corresponding to z which is positive but small does not suggest that H should be rejected in favor of H The rejection of H is appropriate only when x considerably exceeds that is when the z value is pos μ μ . , , , . itive and large In summary the appropriate rejection region based on the test statistic Z rather X has the form Z c ≥ . . ( ) ( The cutoff value c should be chosen to control the probability of type I error at the desired level The required cutoff c is the Z critical value that captures upper tail area under the standard normal curve P Type I error P reject α α α ∴ = =  ) ( ) :...
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This note was uploaded on 03/28/2011 for the course AMS 110 taught by Professor N/a during the Spring '08 term at SUNY Stony Brook.
 Spring '08
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