1. OPLab@im.ntu.edu.tw1 Chi-Square Test and Goodness-of-Fit Testing Ming-Tsung Hsu

2. OPLab@im.ntu.edu.tw 2 Outline Goal of Hypothesis Test Terms & Notation Chi-Square Test Goodness-of-Fit Testing Example

3. OPLab@im.ntu.edu.tw 3 Goal of Hypothesis Test To examine statistical evidence, and to determine whether it supports or contradicts a claim  The life of lamps is more than 10,000 hours  The data are from normal distribution To reduce the directly-relevant data to a “level of suspicion” based purely on the data

4. OPLab@im.ntu.edu.tw 4 Terms & Notation Null Hypothesis (H 0 ) vs. Alternative hypothesis (H 1 or H A )  Type I Error vs. Type II Error Parametric Test vs. Non-Parametric Test Significance level (α) and Critical Region  “Reject H 0 ” vs. “Do not reject H 0 “ Central Limit Theorem  Sampling distribution of the sample mean Test Statistic vs. Table Value  P-value

5. OPLab@im.ntu.edu.tw 5 Null Hypothesis vs. Alternative hypothesis

6. OPLab@im.ntu.edu.tw 6 Type I Error vs. Type II Error Type I error  H 0 is true but reject H 0  Pr(reject H 0 | H 0 ) = α Type II error  H 1 is true but do not reject H 0  Pr(do not reject H 0 | H 1 ) = β

7. OPLab@im.ntu.edu.tw 7 Parametric Test vs. Non-Parametric Test Parametric Test  Parameters of population  Mean test, variance test, etc. Non-Parametric Test  Make no assumptions about the frequency distributions of the variables being assessed  Independent test, distribution test, etc.

8. OPLab@im.ntu.edu.tw 8 Significance level (α) and Critical Region

9. OPLab@im.ntu.edu.tw 9 Central Limit Theorem

10. OPLab@im.ntu.edu.tw 10 Test Statistic vs. Table Value

11. OPLab@im.ntu.edu.tw 11 P-value

12. OPLab@im.ntu.edu.tw 12 Chi-Square Test Non-Parametric Test  T. S. ~χ 2 (ν) Goodness-of-Fit Test  Also known as “Pearson's chi-square test” Independent Test Homogeneity Test

13. OPLab@im.ntu.edu.tw 13 Goodness-of-Fit Testing Used to test if a sample of data came from a population with a specific distribution O i ： Observations of i th group E i ： Expected frequency of i th group k ： Number of groups m: Number of estimated parameters K-1-m: Degree of freedom

14. OPLab@im.ntu.edu.tw 14 Example

15. OPLab@im.ntu.edu.tw 15 Parameter Estimation - λ

16. OPLab@im.ntu.edu.tw 16 Observations and Expected Frequencies IntervalObstF(t) = p(T < t)C.F.Frequency 0 ~ < 11410.21807812.86659 1 ~ < 2.5122.50.45935927.1021914.2356 2.5 ~ < 51850.70770741.7547414.65255 5 ~ < 7.557.50.84197549.676517.921768 7.5 ~ < 105100.91456553.959344.282832 ≧ 10 5 1595.040662 ？!？!

17. OPLab@im.ntu.edu.tw 17 Test Statistic and P-value

18. OPLab@im.ntu.edu.tw 18 Observations and Expected Frequencies - Paper 18 12.8714.2414.657.924.285.04

19. OPLab@im.ntu.edu.tw 19 Re-Grouping IDlowerupperFreq. 10.33.330 23.36.317 36.39.36 4 12.33 5 15.31 6 18.31 7 21.31 ObstF(x)C. F.E. F. 303.30.55632.81232.813 176.30.78846.48613.673 69.30.89953.0206.534 6 ≧ 9.3 1595.980 # of groups = 1+3.322*log(n)