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Hypothesis : Statement about a parameter Hypothesis testing : decision making procedure about the hypothesis Null hypothesis : the main hypothesis H 0 Alternative hypothesis : not H 0, H 1, H A Two-sided alternative hypothesis, uses One-sided alternative hypothesis, uses > or < IME 301 and 312

Hypothesis Testing Process: 1.Read statement of the problem carefully ( * ) 2.Decide on “hypothesis statement”, that is H 0 and H A ( ** ) 3.Check for situations such as: normal distribution, central limit theorem, variance known/unknown, … 4.Usually significance level is given (or confidence level) 5.Calculate “test statistics” such as: Z 0, t 0,, …. 6.Calculate “critical limits” such as: 7.Compare “test statistics” with “critical limit” 8. Conclude “accept or reject H0” IME 301 and 312

FACT H 0 is true H 0 is false Accept no errorType II H 0 error Decision Reject Type I no error H 0 error =Prob(Type I error) = significance level = P(reject H0 | H0 is true) = Prob(Type II error) =P(accept H0 | H0 is false) (1 - ) = power of the test

The P-value is the smallest level of significance that would lead to rejection of the null hypothesis. The application of P-values for decision making: Use test-statistics from hypothesis testing to find P-value. Compare level of significance with P-value. P-value < 0.01 generally leads to rejection of H 0 P-value > 0.1 generally leads to acceptance of H 0 0.01 < P-value < 0.1 need to have significance level to make a decision IME 301 and 312

Test of hypothesis on mean, two-sided No information on population distribution Test statistic: Critical limit: Fail to reject H 0 if or P-value = IME 301 and 312

Test of hypothesis on mean, one-sided No information on population distribution IME 301 and 312 Test statistic: Fail to reject Ho if P-value = or, Fail to reject H0 if

Test of hypothesis on mean, two-sided, variance known population is normal or conditions for central limit theorem holds Test statistic: Critical limit: Fail to reject H 0 if or, p-value = IME 301 and 312

Test of hypothesis on mean, one-sided, variance known population is normal or conditions for central limit theorem holds IME 301 and 312 Test statistic: Fail to reject Ho if P-value = Or, Fail to reject H0 if

Paired samples verses Independent samples For Paired samples calculate the difference in samples and deal with it similar to hypothesis testing on mean of a population. IME 301 and 312

Test of hypothesis on difference of two means variances known, populations are normal or conditions for central limit theorem holds, two-sided, independent samples IME 301 and 312 Null Hypothesis: Test statistics: Alternative Hypotheses Fail to Reject H0 if

Test of hypothesis on variance population is normal or conditions for central limit theorem holds, two-sided Test statistics: Fail to reject H 0 if IME 312

Goodness-of-Fit Test: 1- Select intervals, k=number of intervals 2- Count number of observations in each interval O i 3- Guess the fitted distribution 4- Decide on p = number of parameters of this distribution (if values of parameters are calculated from the sample data), otherwise p = 0. 5- Calculate expected number in each interval e i 6- Calculate Then X 2 has a distribution. 7- Find P-value related to this X 2. If level of significance is greater than this P-value then reject this distribution and try another one. IME 312

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