ECON 10 Final Cheat Sheet

ECON 10 Final Cheat Sheet - Cheat Sheet F-Test Test of...

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Cheat Sheet F-Test: Test of population variance Chi^2 Test: t-test: When looking for a population mean given the sample deviation P-test: Proportion test when there are 2 **First, find the rejection region, and then the test statistic Hypothesis Testing: Chi^2 Test & Probability of making an error! The hypotheses are 10 : 2 0 X H σ and 10 : 2 1 X H σ and are tested at the 5% significance level using a sample size of 21 to estimate the population variance. If the population variance 75045 . 32 2 = X σ , what is the probability of making a Type II error ? So, .05 with 20 degrees of freedom STEP 1: Rejection region : 4104 . 31 2 20 , 05 . 0 2 = χ χ ****Look at the chart**** 7052 . 15 20 4104 . 31 10 ) 1 ( ) 1 ( 2 2 2 2 2 2 2 = = - = - = L s n s s n χ σ σ χ ***Change formula because you have chi^2 from above. Or TI89 STEP 2: FIND THR TEST RR: Rejection region: 7052 . 15 2 L s 025 . ) 59083 . 9 P( ) 32.75045 15.7052 20 ) 1 ( P( ) ( ) 32.75045 7052 . 15 ( ) ( ) 32.75045 ( ) true is H Null Rejecting Not ( ) ( 2 2 2 2 2 H 2 2 H 2 2 1 0 0 = < = < - = = = < = = < = = χ σ χ σ σ s n II Type P s P II Type P s s P P II Type P L --------------------------------------------------------------------------------------------------------------------------------- Hypothesis Testing – P-test The hypotheses that 1 . 0 : 2 1 0 - p p H and 1 . 0 : 2 1 1 < - p p H are tested at the 1% significance level using a sample proportion from population 1 50 . 0 ˆ 1 = p and a sample proportion from population 2 60 . 0 ˆ 2 = p . Assume the distribution of the difference in the sample proportions ( 2 1 ˆ ˆ p p - ) is normally distributed and the standard deviation of the difference of the sample proportion is 10 . 0 2 1 ˆ ˆ = - p p σ . If 62 . 0 1 = p and 56 . 0 2 = p what happens in the problem? 0 . 2 10 . 0 20 . 0 10 . 0 1 . 0 ) 60 . 0 50 . 0 ( ) ( ) ˆ ˆ ( ) ˆ ˆ ( 2 1 2 1 2 1 ˆ ˆ 2 1 2 1 ˆ ˆ ˆ ˆ 2 1 - = - = - - = - - - = - - = - - - p p p p p p p p p p p p Z σ σ μ Rejection Region: 33 . 2 01 . 0 - = - z z ****GO TO THE P CHART!**** (.001 comes from sig.) If 62 . 0 1 = p and 56 . 0 2 = p then 10 . 0 06 . 0 56 . 0 62 . 0 2 1 < = - = - p p and the alternative is true. We do not reject the null when the alternative is true, we therefore have made a Type II Error .
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--------------------------------------------------------------------------------------------------------------------------------- Hypothesis testing – F-TEST The hypothesis that 2 2 0 2 1 10 : X X H σ σ and 2 2 1 2 1 10 : X X H σ σ is tested at the 5% significance level, where the sample variance of population 1 500 2 1 = X s is obtained from a sample of size 4 and the sample variance of population 2 10 2 2 = X s is obtained from a sample of size 6. If 400 2 1 = X σ and 50 2 2 = X σ which of the following statements is correct. Testing 2 2 0 2 1 10 : X X H σ σ and 2 2 0 2 1 10 : X X H σ σ is tested at the 5% significance level: Rejection region: F F 0.05, 3, 5 ≥ 5.41 **By looking at the chart Test statistics: 0 . 5 10 1 10 500 2 1 2 2 2 2 2 1 = = = σ σ s s F The test statistic does not fall in the rejection region therefore we do not reject the null hypothesis. If 400 2 1 = X σ and 50 2 2 = X σ , then 8 50 400 2 2 2 1 = = X X σ σ and the null is true. We do not reject the null hypothesis and have not committed a Type I or Type II error. --------------------------------------------------------------------------------------------------------------------------------- Hypothesis Testing – u1-u2 (Because of ux) & Type I Error!
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