1. EGR 252 S09 Ch.10 Part 3 Slide 1 Statistical Hypothesis Testing - Part 3  A statistical hypothesis is an assertion concerning one or more populations.  In statistics, a hypothesis test is conducted on a set of two mutually exclusive statements: H 0 : null hypothesis H 1 : alternate hypothesis  New test statistic of interest:

2. EGR 252 S09 Ch.10 Part 3 Slide 2 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. Example: H 0 : The data come from a normal distribution. H 1 : The data do not come from a normal distribution.

3. EGR 252 S09 Ch.10 Part 3 Slide 3 Goodness of Fit Tests (cont.)  Test statistic is χ 2  Draw the picture  Determine the critical value for goodness of fit test χ 2 with parameters α, ν = k – 1  Calculate χ 2 from the sample  Compare χ 2 calc to χ 2 crit  Make a decision about H 0  State your conclusion.  Discussion: Look at Table 10.4 in text.

4. EGR 252 S09 Ch.10 Part 3 Slide 4 Tests of Independence  Example:Choice of pension plan.  Hypotheses H 0 : Pension Plan Choice and Worker Type are independent H 1 : Pension Plan Choice and Worker Type are not independent 1. Develop a Contingency Table Worker Type Pension Plan Total #1#2#3 Salaried16014040340 Hourly4060 160 Total200 100500

5. EGR 252 S09 Ch.10 Part 3 Slide 5 Worker vs. Pension Plan Example 2. Calculate expected probabilities P(#1 ∩ S) = ____________E(#1 ∩ S) = __________ P(#1 ∩ H) = ____________E(#1 ∩ H) = __________ (etc.) Worker Type Pension Plan Total #1#2#3 Salaried16014040340 Hourly4060 160 Total200 100500 #1#2#3 S (exp.) H (exp.)

6. EGR 252 S09 Ch.10 Part 3 Slide 6 Hypotheses 3.Define Hypotheses H 0 : the categories (worker & plan) are independent H 1 : the categories are not independent 4. Calculate the sample-based statistic (160-136)^2/136 + (140-136)^2/136 + (40-68)^2/68 + (40-64)^2/64 + (60-64)^2/64 + (60-32)^2/32 = 49.63

7. EGR 252 S09 Ch.10 Part 3 Slide 7 The Chi-Squared Test of Independence 5. Compare to the critical statistic for a test of independence, χ 2 α, r where r = (a – 1)(b – 1) a = # of columns b = # of rows For our example, let’s use α = 0.01 _ χ 2 0.01,2 _ = 9.210 (from Table A.5, pp 756) Comparison: χ2 calc> χ2 crit Decision: Reject the null hypothesis Conclusion: W orker and plan are not independent.

8. EGR 252 S09 Ch.10 Part 3 Slide 8