1. Goodness-of-fit (GOF) Tests Testing for the distribution of the underlying population H 0 : The sample data came from a specified distribution. H 1 : The sample data did not come from the specified distribution.

2. 9-7 The Chi-Square GOF Test Assume there is a sample of size n from a population whose probability distribution is unknown. Let O i be the observed frequency in the ith class interval. Let E i be the expected frequency in the ith class interval. The test statistic is chi-square with df = k – p – 1 where p = number of estimated parameters

3. Chi-Square GOF Test where k = number of classes O i = observed number in the i th class E i = expected number in the i th class = n p i n = sample size p i = F(a i ) - F(a i-1 ) = probability of a failure occurring in the i th class if H 0 is true with df = k - 1 - number of estimated parameters Hypothesized distribution i th class is defined by [a i-1, a i ) with a 0 = 0

4. Example 9-12

5. More of Example 9-12

6. Much More of Example 9-12

7. Still Example 9-12

8. Yes, Example 9-12

9. The last of Example 9-12

10. Is this Normal? Three years of daily high temperatures in Dayton during the month of August yields the following data: Enter the raw data Stats 12345678910sample size93 177818581888679878391mean83.806 2879489 808384809088variance20.658 3857685 847784808384std dev4.545 4937982908683 8683 median84 585828882 83898676 1st quartile80 694798778 7684 3rd quartile84 78376 838281928679 interquartile rg4 88880 888486847773 Mimimum73 9808385888590847985 Maximum94 10898086858991888985 Range21 Skewness0.0043 Kurtosis-0.4465

11. The observed count Sturges rule: k = 1+3.3*LOG(90,10) = 7.495 ; 7 or 8 classes BinFrequency 751 7812 8115 8423 8720 9016 934 962 More0

12. The expected count H 0 : High temperatures during August in Dayton are normally distributed H 1 : High temperatures in August do not have a normal distribution mean83.806 variance20.658 std dev4.545

13. The Chi-Sq Statistic intervalUpper bndobservedexpected(O-E)^2/E 17512.44590.854747 278126.94713.675174 3811515.61470.024199 4842323.11980.000621 5872022.48740.275139 6901614.35920.187491 79346.02640.681385 89621.66470.067535 9infinity00.3348 sum93 6.101091 df = 9 – 2 – 1 = 6 Cannot reject the null hypothesis that the sample came from a normal distribution

14. combine cells intervalupperobservedexpected(O-E)^2/E 178139.3931.385122 2811515.61470.024199 3842323.11980.000621 4872022.48740.275139 5901614.35920.187491 6infinity68.02590.511378 sum93 2.38395 intervalUpper bndobservedexpected(O-E)^2/E 17512.44590.854747 278126.94713.675174 3811515.61470.024199 4842323.11980.000621 5872022.48740.275139 6901614.35920.187491 79346.02640.681385 89621.66470.067535 9infinity00.3348 sum93 6.101091 df = 6 – 2 – 1 = 3