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slide4 - Hypothesis testing example An automotive company...

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1 Hypothesis testing: example  An automotive company claims that its new motorcycle  produces 145 hp at the rear wheel. It is known that the  standard deviation for the whole population is equal to 3  hp.  Let’s take several size 10 samples and try to find out  whether the statement can be trusted. Given a value of the sample mean suggest a criterion  according to which the claim (hypothesis) can be  rejected. How easy is it to reject a true claim?
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2 Definitions Critical region : range of values of the  test statistic for which the null  hypothesis is rejected Acceptance region : range of values of  the statistic for which the null hypothesis  is not rejected Critical values : boundaries between the  critical and acceptance regions
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3 More Definitions Type I error : rejecting the null hypothesis  when it is true Type II error : failing to reject the null  hypothesis when it is false Significance level : probability of type I error Power of the test : probability of (correctly!)  rejecting the null hypothesis when it is false:  (power)=1-(type II error probability)
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4 Hypothesis Testing α  = P(Type I error) = P(reject H 0  | H 0  is true) β  = P(Type II error) = P(accept H 0  | H 0  is false) Power = P(reject H 0 | H 0  is false) = 1 -  β Decision H 0  is true H 0  is false Fail to reject H 0 No error Type II error Reject H 0 Type I error No error
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5 Statistical inference: sequence of  steps Identify the relevant test statistic (usually  “standardized” version of a good point  estimator) Determine the sampling distribution when the  null hypothesis is true Find the corresponding critical values (from  some table) Get a random sample and conduct the test.
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6 Hypothesis Tests on the Mean H 0 μ  =  μ 0   H 1 μ     μ 0   n X Z 0 0 σ μ - = 2 / 0 2 / 0 2 / 0 2 / 0 0 z Z z if H reject to Fail z Z or z Z if H Reject α α α α - <
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7 Hypothesis Tests (one side) H 0 μ  =  μ 0   H 1 μ  >  μ 0 H 0 μ  =  μ 0   H 1 μ  <  μ 0 α z Z if H Reject 0 0 α z Z if H Reject 0 0 - <
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8 Example: two-sided test Suppose we take a random sample of  n =25 and  obtains an average horsepower of 143.2. The claimed mean horsepower is 145, and the  standard deviation is known to be 3 hp. We decide to specify a type I error probability  (significance level) of 0.05. Should we doubt the claim?  What would the result be in the one-sided test  case?
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9 P-value The P-value is the smallest level of significance that  would lead to rejection of the null hypothesis H 0  with the  given data.
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