Test on equality of variances Given two independent

Test on equality of variances Given two independent populations, X 1 ~N(μ 1, σ12) and X 2 ~N(μ 2, σ22). Before testing for equality of means, you want to test the hypothesis that the variances of the populations are equal, i. e. , σ12 =σ22. The Hypotheses are … H 0: ________ H 1: ________ 1 and the test statistic is ________ ETM 620 - 09 U

Example 11. 18, pg. n 1 = 8, s 12 = 3. 89 n 2 = 8, s 22 = 4. 02 Picture Hypotheses: Analysis: F 0 = _______ F____ , ____= ____________ Decision: Conclusion: 2 ETM 620 - 09 U

Goodness-of-Fit Tests Procedures for confirming or refuting hypotheses about the distributions of random variables. Hypotheses: H 0: The population follows a particular distribution. H 1: The population does not follow the distribution. Examples: H 0: The data come from a normal distribution. H 1: The data do not come from a normal distribution. 3 ETM 620 - 09 U

Goodness of Fit Tests (cont. ) Test statistic is χ2 Draw the picture Determine the critical value χ2 with parameters α, ν = k – p – 1 Calculate χ2 from the sample Compare χ2 calc to χ2 crit Make a decision about H 0 State your conclusion 4 ETM 620 - 09 U

Example: Problem 11. 48, pg. 318 1. Estimate the mean, λ 2. Find the expected frequencies, Ei 3. If any cell has < 5 observations, combine cells. 4. Compute 5. Compare to Χ 2α, k-p-1 Number of Defects, i Number of Wafers with i Defects Ei 0 4. 00 2. 76 1 13. 00 13. 72 2 34. 00 34. 15 3 56. 00 56. 66 4 70. 00 70. 50 5 70. 00 70. 18 6 58. 00 58. 22 7 42. 00 41. 40 8 25. 00 25. 76 9 15. 00 14. 25 10 9. 00 7. 09 11 3. 00 3. 21 12 1. 00 1. 33 total = 400 λ= 4. 9775

Tests of Independence Hypotheses H 0: independence H 1: not independent Example Choice of pension plan. 1. Develop a Contingency Table Pension Plan 6 Worker Type #1 #2 #3 Total Salaried 160 140 40 340 Hourly 40 60 60 160 Total 200 100 500 ETM 620 - 09 U

Example Pension Plan Worker Type #1 #2 #3 Total Salaried 160 140 40 340 Hourly 40 60 60 160 Total 200 100 500 2. Calculate expected probabilities P(#1 ∩ S) = ________ E(#1 ∩ S) = _______ P(#1 ∩ H) = ________ E(#1 ∩ H) = _______ (etc. ) #1 #2 #3 S (exp. ) 7 H (exp. ) ETM 620 - 09 U

Hypotheses Define Hypotheses H 0: the categories (worker & plan) are independent H 1: the categories are not independent 4. Calculate the sample-based statistic 3. = ____________________ = ______ 8 ETM 620 - 09 U

The Test 5. Compare to the critical statistic, χ2α, r where r = (a – 1)(b – 1) for our example, say α = 0. 01 χ2_____ = ______ Decision: Conclusion: 9 ETM 620 - 09 U