1. Prof. Dr. Ahmed Farouk Abdul Moneim

2. 1) Uniform Didtribution 2) Poisson’s Distribution 3) Binomial Distribution 4) Geometric Distribution 5) Negative Binomial Distribution 6) Hyper Geometric Distribution

3. Example 1 Ministry of Health decided to locate Ambulance stations along the road Cairo-Alexandria. One of the important factors in defining the number of the stations is the Number of Accidents per day on that road. A statistic of 100 days has been collected. The statistic reveals that the Number of Accident per day along the road is random ranging from 0 to 6 The following observed frequency of accidents are given in the following table: X 0123456 Observed Frequency 6132520121311 Is POISSON’S Distribution Appropriate for the data given above? _______________________________________________________________ Step 2 Find estimators for the parameters of the Hypothesized Distribution from SAMPLE 2.1 Write down the Hypothesized distribution (POISSON’s Distribution) Step 1 Statement of Test: Ho : The Poisson’s Distribution is considered Appropriate for the data (Observed Frequency from Data - Expected Frequency according to Distribution)^2 = 0 H1 : The Poisson’s Distribution is NOT considered Appropriate for the data (Observed Frequency from Data - Expected Frequency according to Distribution)^2 >> 0

4. We have only ONE parameter μ 2.2 Estimator for the parameter μ is evaluated as follows: Step 3 Find The EXPECTED FREQUENCY based on the Hypothesized Distribution Expected Frequency (X = x) = Sample Size * P(X = x ) X 0123456 or more Sum P(X =x) 0.049780.14940.2204 0.1680.10080.08391 Expected Frequency 4.97814.9422.04 16.810.088.39100 Sample size = 100 days X 0123456 Observed Frequency 6132520121311

5. Step 4 Test STATISTIC: X Observed Frequency O Expected Frequency E 064.9780.2095 11314.940.25097 22522.040.30076 32022.040.25799 41216.81.3729 51310.080.8446 6118.390.8106 Sum =4.0475 X0123456 or moreSum P(X =x)0.049780.14940.2204 0.1680.10080.08391 Expected Frequency 4.97814.9422.04 16.810.088.39100 X0123456 Observed Frequency 6132520121311

6. Step 5: Define The Zone of ACCEPTANCE α Take α = 0.05 K = Number of data points of X P = Number of the parameters of the distribution From tables or Excel Accept Step 6 : Perform the hypothesis test by Comparing with = 4.0475 <<< = 11.07, Therefore we ACCEPT Ho Since we ACCEPT Ho, then the POISSON’S distribution is considered APPROPRIATE distribution for the variable X Prof. Dr. Ahmed Farouk Abdul Moneim

7. Example 2 The number of under filled bottles from a filling operation in a carton of 24 bottles is random variable in the range from 0 to 3. 75 cartons are inspected and the following observation are recorded: X 0123 Observed Frequency 3923121 Is the Binomial Distribution an Appropriate model for the variable X? Step 2 Find estimators for the parameters of the Hypothesized Distribution from SAMPLE 2.1 Write down the Hypothesized distribution (BINOMIAL Distribution) Step 1 Statement of Test: Ho : The Binomial Distribution is considered Appropriate for the data (Observed Frequency from Data - Expected Frequency according to Distribution)^2 = 0 H1 : The Binomial Distribution is NOT considered Appropriate for the data (Observed Frequency from Data - Expected Frequency according to Distribution)^2 >> 0 We have only ONE parameter p since N is given N = 24 bottles 2.2 Estimator for the parameter p is defined as follows:

8. Step 3 Find The EXPECTED FREQUENCY based on the Hypothesized Distribution Expected Frequency = Sample Size * P(X = x ) X 0123Sum P(X =x) 0.5080.348850.114710.028121 Expected Frequency 38.1226.16358.6042.1975 N = 24 bottlesSample size = 75 cartons Step 4 Test STATISTIC: X Observed Frequency O Expected Frequency E 03938.120.0201 12326.16350.3825 2128.6041.34 312.190.583 Sum =2.326 O = Observed Frequency E = Expected Frequency X 0123 Observed Frequency 3923121

9. Step 5: Define The Zone of ACCEPTANCE α Take α = 0.05 K = Number of data points of X P = Number of the parameters of the distribution From tables or Excel Step 6 : Perform the hypothesis test by Comparing with = 2.326 << = 5.9915, Therefore we ACCEPT Ho Since we ACCEPT Ho, then the BINOMIAL distribution is considered APPROPRIATE distribution for the variable X Accept

10. Prof. Dr. Ahmed Farouk Abdul Moneim 1) Normal Distribution 2) Exponential Distribution 3) Erlang Distribution 4) Weibull Distribution 5) Chi-Square Distribution 6) Beta Distribution 7) Lognormal Distribution 8) Rayleigh Distribution 9) Gamma Distribution

11. Example 1 A sample of 24 microprocessors are put on life test. The time to failure of these 24 Microprocessors are given in the following table in hours: 94201929867144055752675268774721159 10205131115958835385891149131141311037 a) Test the hypothesis that the NORMAL Distribution is the APPROPRIATE distribution of the time to failure as given in the sample above. b) If INAPPROPRIATE, test the Appropriateness of EXPONENTIAL Distribution Test for NORMAL DISTRIBUTION Step 1 Statement of Test: Ho : The NORMAL Distribution is considered Appropriate for the data (Observed Frequency from Data - Expected Frequency according to Distribution)^2 = 0 H1 : The NORMAL Distribution is NOT considered Appropriate for the data (Observed Frequency from Data - Expected Frequency according to Distribution)^2 >> 0 Prof. Dr. Ahmed Farouk Abdul Moneim

12. Step 2 Find Estimators for Parameters of Hypothesized Distribution from SAMPLE 2.1 Write down the Hypothesized distribution (Normal Distribution) Z We have TWO parameters μ, σ 2.2 Estimator for the parameter μ and Estimator S for parameter σ are evaluated Step 3 Find The EXPECTED FREQUENCY based on the Hypothesized Distribution For CONTINUOUS Random Variables, The AREA under the Probability curve which is equal to ONE, will be subdivided into EQUAL M SEGMENTS The Number of Equal Segments = where, N = Number of data points (as in Histogram) X1X2X3X4 1 4 32 5 Equal areas The number of Data points N will be EQUALLY Divided over the M Segments Expected Frequency is Constant for all Segments Expected Frequency = N / M = 24 / 5 = 4.8 Prof. Dr. Ahmed Farouk Abdul Moneim

13. Find x1, x2, x3 and x4 REMEMBER ! Area to the Left Z from tables x 1 0.2-0.841 326.6 2 0.4-0.253 419.51 3 0.60.253 499.53 4 0.80.841 592.44 X1X2X3X4 1 4 3 2 5 Step 4 Find The OBSERVED FREQUENCY based on the Sample Data 14316794111131141267298405472513 52653857558959587788310201037114911592019 SORT the Data of the Sample in an Ascending Order SegmentNumber of Points in the Segment = Observed Frequency X <= 326.6 9 326.6 < X <= 419.51 1 419.51 < X<= 499.53 1 499.93< X <= 592.44 5 X > 592.44 8 Prof. Dr. Ahmed Farouk Abdul Moneim

14. Step 5 TEST STATISTIC Segment Observed Frequency O Expected Frequency E X <= 326.694.8 3.68 326.6 < X <= 419.5114.8 3.01 419.51 < X<= 499.5314.8 3.01 499.93< X <= 592.4454.8 0.01 X > 592.4484.8 2.13 Sum =11.833 Step 6: Define The Zone of ACCEPTANCE P-Value From Tables P-Value(11.833, 2) = 0.0027 K = Number of data points of X P = Number of the parameters of the distribution

15. Step 7 : Perform the hypothesis test judging from the rule of P-Value Since we REJECT Ho, then the NORMAL distribution is considered INAPPROPRIATE distribution for the variable X If P-Value ≤ 0.01 REJECT Ho absolutely If P-Value ≥ 0.1 ACCEPT Ho absolutely Otherwise it is up to concerned parties P-Value = 0.0027 <<< 0.01 Therefore, We REJECT Ho

16. TESTING THE APPROPRIATENESS of the EXPONENTIAL Distribution for the above Data Step 1 Statement of Test: Ho : The EXPONENTIAL Distribution is considered APPROPRIATE for the data (Observed Frequency from Data - Expected Frequency according to Distribution)^2 = 0 H1 : The EXPONENTIAL Distribution is considered INAPPROPRIATE for the data (Observed Frequency from Data - Expected Frequency according to Distribution)^2 >> 0 Step 2 Find Estimators for Parameters of Hypothesized Distribution from SAMPLE 2.1 Write down the Hypothesized Exponential Distribution) We have ONE parameters μ 2.2 Estimator for the parameter is evaluated Step 3 Find The EXPECTED FREQUENCY based on the Hypothesized Distribution X A As already explained above: Expected Frequency is Constant for all Segments Expected Frequency = N / M = 24 / 5 = 4.8 1/μ

17. Find x1, x2, x3 and x4 Area to the Left X from Formula (1) 1 0.2 125.63 2 0.4 287.6 3 0.6 515.87 4 0.8 906.1 Step 4 Find The OBSERVED FREQUENCY based on the Sample Data 14316794111131141267298405472513 52653857558959587788310201037114911592019 SORT the Data of the Sample in an Ascending Order SegmentNumber of Points in the Segment = Observed Frequency X <= 125.63 5 125.63 < X <= 287.61 3 287.61 < X<= 515.87 4 515.87< X <= 906.1 7 X > 906.1 5 X1X2X3X4 1 4 32 5 1/μ Prof. Dr. Ahmed Farouk Abdul Moneim

18. Step 5 TEST STATISTIC Segment Observed Frequency O Expected Frequency E X <= 125.63 5 4.8 0.008 125.63 < X <= 287.61 3 4.8 0.675 287.61 < X<= 515.87 4 4.8 0.134 515.87< X <= 906.1 7 4.8 1.008 X > 906.1 5 4.8 0.008 Sum = 1.83 Step 6: Define The Zone of ACCEPTANCE From Tables P-Value(1.83, 3) = 0.6084 K = Number of Classes P = Number of the parameters of the distribution P-Value Prof. Dr. Ahmed Farouk Abdul Moneim

19. Step 7 : Perform the hypothesis test judging from the rule of P-Value Since we ACCEPT Ho, then the EXPONENTIAL distribution is considered APPROPRIATE distribution for the variable X If P-Value ≤ 0.01 REJECT Ho absolutely If P-Value ≥ 0.1 ACCEPT Ho absolutely Otherwise it is up to concerned parties P-Value = 0.6084 >>> 0.1 Therefore, We ACCEPT Ho