ࡱ> )+(@ bjbjFF (>,,         8 Tn L  $sR&  &   ;000 R   0 00  0 @Ѱz  j0dQ00~ 0     0l 0 && &  HYPOTHESIS TEST AND CONFIDENCE INTERVALS FOR 1 AND 2 SAMPLE MEANS AND PROPORTIONS and VARIANCES (standard deviations) Basic Formula: z(or t) = (non z(or t) of interest - mean )/ SD Comment: for hypothesis test if given a level of significance first set up the rejection area(s) and find the edge(s) of it by using the z(or t) table Comment: for hypothesis tests the z(or t) from the data is (column 3 column 4)/column 5 Comment: the confidence interval for each case below is column 3 +/- column 2 * column 5 Comment: in any of the cases below if the population standard deviation(s) are known then use them and z Comment: sample size needed for CI for mean is  EMBED Equation.3 , sample size for CI for proportion is  EMBED Equation.3 (use  EMBED Equation.3 to guarantee sample size is large enough, use  EMBED Equation.3 in place of  EMBED Equation.3 to get reasonable estimate) Situationz/tNon z (or t) of interestMeanStandard Deviation or Standard Error1 sample meant, df = n-1sample meanpopulation mean in Ho EMBED Equation.3 = EMBED Equation.3 Difference of means from 2 dependent samples (a.k.a. matched pairs)t, df = n-1sample mean of the differences subtracted in appropriate orderpopulation difference mean in Ho (often 0) EMBED Equation.3 = EMBED Equation.3  Where s is the s.d. of the differences. Difference of means from 2 independent samplest, df = min of the sample sizes - 1the difference of the two sample means subtracted in appropriate orderdifference of the population means in Ho (often 0) EMBED Equation.3 Difference of means from 2 independent samples with the assumption that the population standard deviations are equalt, df = EMBED Equation.3 the difference of the two sample means subtracted in appropriate orderdifference of the population means in Ho (often 0) EMBED Equation.3  where  EMBED Equation.3 = EMBED Equation.3 1 sample proportionzsample proportion, p=number of successes / npopulation proportion, p, in HoHT:  EMBED Equation.3  CI:  EMBED Equation.3 Difference of proportions (percentages, or probabilities of success) from 2 sampleszthe difference of the two sample proportions subtracted in appropriate orderdifference in population proportions in Ho (often 0) (0 and non 0 differences have different standard deviations see ( HT(0 case):  EMBED Equation.3 where  EMBED Equation.3  EMBED Equation.3  CI & HT(non 0 case):  EMBED Equation.3  Confidence interval for population variance: Use  EMBED Equation.3  , get two values from the  EMBED Equation.3 table and solve for  EMBED Equation.3 twice. (Take square root if you want  EMBED Equation.3 ) Hypothesis test for population variance: Get critical value(s) from  EMBED Equation.3 table, and use  EMBED Equation.3 where  EMBED Equation.3 is from Ho. Hypothesis test for ratio of two variances: Use  EMBED Equation.3 maker sure the top is bigger than the bottom. Keep track of the two different dfs. Use the F-table for the critical value(s). Note that making F(data)>1 even if you have two critical values as in a two-tail test, only the right hand one matters. 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