Chi-Square Test and Goodness-of-Fit Testing Ming-Tsung Hsu

2 Outline Goal of Hypothesis Test Terms & Notation Chi-Square Test Goodness-of-Fit Testing Example

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 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 Null Hypothesis vs. Alternative hypothesis

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 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 Significance level (α) and Critical Region

9 Central Limit Theorem

10 Test Statistic vs. Table Value

11 P-value

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 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 Example

15 Parameter Estimation - λ

16 Observations and Expected Frequencies IntervalObstF(t) = p(T < t)C.F.Frequency 0 ~ < ~ < ~ < ~ < ~ < ≧ ？!？!

17 Test Statistic and P-value

18 Observations and Expected Frequencies - Paper

19 Re-Grouping IDlowerupperFreq ObstF(x)C. F.E. F ≧ # of groups = *log(n)