1. Analyze Statistic by Using SPSS 2 nd Day 1Fadwa Flemban

2. الإعجاز الإحصائي للقرآن الإعجاز الإحصائي للقرآن يعني أننا عاجزون عن تأليف كتاب فيه نظام رقمي دقيق لتكرار الكلمات مثل القرآن. } إِنَّ هَذِهِ تَذْكِرَةٌ فَمَنْ شَاءَ اتَّخَذَ إِلَى رَبِّهِ سَبِيلاً} هذه الآية الكريمة نجدها في موضعين فقط من القرآن : 1 ـ { إن هذه تذكرة فمن شاء اتخذ إلى ربه سبيلاً } [ المزمل : 19 [ 2 ـ { إن هذه تذكرة فمن شاء اتخذ إلى ربه سبيلاً } [ الإنسان : 29[ 1 ـ لماذا تكررت هذه الآية مرتين في القرآن ؟ 2 ـ لماذا كان رقم الآيتين : 19 ـ 29 ؟ 3 ـ لماذا كان تسلسل سورة المزمل قبل سورة الإنسان ؟ سوف نجيب على هذه الأسئلة وغيرها بلغة الأرقام ، فتكرار هذه الآية في القرآن مرتين له حكمة ، وقد اختار اللّه تعالى لهذه الآية رقمين 19 و 29 وذلك لحكمة أيضاً ، ويمكن استنتاج جزء من هذه الحكمة بالاعتماد على الرقم 7. والفكرة الأساسية في هذا البحث تعتمد على صَفّ أرقام الآيات بجانب بعضها وذلك حسب تسلسل هذه الآيات في القرآن (وليس جَمع الأرقام ) وعندما نصُفّ أرقام الآيتين 19ـ 29 ينتج عدد جديد هو 2919 هذا العدد يُقرأ : ألفان وتسعمئة وتسعة عشر ، وهو من مضاعفات الرقم 7 ، أي يقبل القسمة تماماً على 7 من دون باقٍ : 417 × 7 = 2919 2Fadwa Flemban

3. Probability Distributions بعض التوزيعات الاحتمالية 1.The normal distribution 2.The Standard normal distribution 3.T distribution 4.Chi-square distribution 3Fadwa Flemban

4. (1)Normal Distribution The normal distribution is a probability distribution that associates the normal random variable X with a cumulative probability. The normal distribution is defined by the following equation:probability distribution normal random variable cumulative probability Y = [ 1/σ * sqrt(2π) ] * e -(x - μ)2/2σ2 the sampling distribution of a statistic will follow normal distribution, as long as : 1- the sample size is sufficiently large.sampling distribution normal distribution 2-we know the standard deviation of the population. Fadwa Flemban4

5. The Curve of Normal Distribution: Fadwa Flemban5 Find P(Z > a)? P(Z > a) = 1 - P(Z < a) Find P(a < Z < b)? P(Z < b) - P(Z < a) = P(Z < z)

6. Difference between these curves: The curve on the left is shorter and wider than the curve on the right, because the curve on the left has a bigger standard deviation. Fadwa Flemban6

7. (2)The standard normal distribution The standard normal distribution is a special case of the normal distribution.normal distribution the following equation: z = (X - μ) / σ Z ~ N ( μ, σ ) in standard normal distribution: Z ~ N ( 0, 1 ) Fadwa Flemban7

8. Standard Normal Distribution Table Find the cumulative probability of a z-score equal to -1.31? Fadwa Flemban8 z0.000.010.020.030.040.050.060.070.08 -3.00.0013 0.0012 0.0011 0.0010... -1.40.08080.07930.07780.07640.07490.07350.07220.07080.0694 -1.30.09680.09510.09340.09180.09010.08850.08690.08530.0838 -1.20.11510.11310.11120.10930.10750.10560.10380.10200.1003... 3.00.9987 0.9988 0.9989 0.9990 The table shows that the probability that a standard normal random variable will be less than -1.31, that is, P(Z < -1.31) = 0.0951.

9. قال تعالى : ( اللَّهُ الَّذِي خَلَقَكُم مِّن ضَعْفٍ ثُمَّ جَعَلَ مِن بَعْدِ ضَعْفٍ قُوَّةً ثُمَّ جَعَلَ مِن بَعْدِ قُوَّةٍ ضَعْفاً وَشَيْبَةً يَخْلُقُ مَا يَشَاءُ وَهُوَ الْعَلِيمُ الْقَدِيرُ ) [ الروم : 54] Fadwa Flemban9 ضعف قـوة ضعف

10. (3) t Distribution sample sizes are sometimes small, and often we do not know the standard deviation of the population. When either of these problems occur, statisticians rely on the distribution of the t statistic. The equation : t = [ x - μ ] / [ s / sqrt( n ) ] Degrees of Freedom: the sample size minus one (n-1, α). Fadwa Flemban10

11. Fadwa Flemban11 Curve of t Distribution

12. (4)Chi-Square Distribution using the following equation: Χ 2 = [ ( n - 1 ) * s 2 ] / σ 2 degrees of freedom Χ 2 : n - 1 Fadwa Flemban12

13. Curve of Chi-Square As the degrees of freedom increase, the chi-square curve approaches a normal distribution. Fadwa Flemban13

14. Fadwa Flemban14 Inferential Statistics الاحصاء الاستدلالي

15. The First Topic in Inferential Statistics Estimation Fadwa Flemban15

16. Estimation التقدير When a parameter is being estimated, the estimate can be either a single number or it can be a range of scores.parameter When the estimate is a single number, the estimate is called a "point estimate" When the estimate is a range of scores, the estimate is called an interval estimate. Confidence interval are used for interval estimates. 16Fadwa Flemban the population….. estimate of Sample ….. Mean µ mean M or variance σ²variance s²

17. Estimation by SPSS Analyze  Descriptive Statistics  Explore Fadwa Flemban17

18. Fadwa Flemban18 Statistics

19. Output: Fadwa Flemban19 105.42 < µ < 118.98 µ = 112.2 σ ² = 89.73 All the estimations at Confidence interval 95%

20. The Second Topic in Inferential Statistics Testing Hypotheses Fadwa Flemban20

21. Fadwa Flemban21 Inferential Statistics الاحصاء الاستدلالي

22. Tests Concerning a Single Mean اختبارات الفروض حول متوسط المجتمع Single mean µ = µₒ n>30 Z Distribution n ≤30 T Distribution Fadwa Flemban22

23. Summary of Computational Steps  Specify the null hypothesis and an alternative hypothesis.  Compute M = ΣX/N.  Compute, if σ unknown compute  Compute, if n<30 compute  where M is the sample mean and µ is the hypothesized value of the population.  Use a z table to determine p from z, orz table Use a t table to compute p from t and df (df=N-1).t table Fadwa Flemban23

24. Testing Hypotheses اختبارات الفروض Consider an experiment designed to test the null hypothesis that µ = 10. The test would be conducted with the following formula: where M (the statistic) is the sample mean.samplemean Fadwa Flemban24

25. Tests Concerning a Single Mean Example n > 30 A random sample of 100 deaths in the Philippines last year showed an average life span of 69.3 years. Assuming a population standard deviation of 7.8 years. does this seem to indicate that the life span today is lesser than 70 years? Use a 0.01 level of significance? Fadwa Flemban25

26. We will solve this testing problem in 5 steps: 1. H0:  = 70 years. H1:  < 70 years. 2.Use a = 0.01. 3. Since we are testing the mean life span and the population standard deviation is known (  / = 0.78). Using normal distribution, 4. At 0.01 level of significance, we reject if and only if Z < z 0.01= -2.33 5. Based from the rejection region (and critical value – 2.33), we see that the z value is outside the rejection region (or – 0.90 is greater than – 2.33). Thus, we do not reject the null hypothesis and conclude that the life expectancy of Filipinos is 70 years. Solution: Z = -0.90 -2.33 Rejection Region 26Fadwa Flemban

27. Note The previous Example We cannot use SPSS because: 1- the standard deviation is known. 2- and we haven’t data. Fadwa Flemban27

28. Tests Concerning a Single Mean Example n ≤ 30 Fadwa Flemban28 A teacher of Arabic language was assumed that the mean of students' degree is 68, drew a sample from the students' degree as follows (n=30): 8085756555524433 30254580955030 9588907772756040 57555248848778 Is teacher's claim is true at 0.01 level of significance ? A teacher of Arabic language was assumed that the mean of students' degree is 68, drew a sample from the students' degree as follows (n=30): 8085756555524433 30254580955030 9588907772756040 57555248848778 Is teacher's claim is true at 0.01 level of significance ?

29. H 0 : µ = 68 H 1: µ  68 Fadwa Flemban29 SPSS Solution:

30. Fadwa Flemban30 Analyze  compare mean  One-sample test

31. Fadwa Flemban31 Option

32. Fadwa Flemban32 Output (P-value) 0.240 > 0.01 (α) don't reject Hₒ the teacher's claim is true. (P-value) 0.240 > 0.01 (α) don't reject Hₒ the teacher's claim is true.

33. Fadwa Flemban33 Tests of Differences between Means اختبارات الفروض حول متوسطي مجتمعين Two means µ1 = µ2 Two Independent Samples Independent samples t-test Two Dependent Samples Paired samples t-test

34. Tests of Differences between Means, Independent Samples Example: If we have the following marks for students (male & female): Is there a difference between the mean of marks at level of significant 0.05? Fadwa Flemban34 7577305084805225303344525565758580 Female 73788795484555574060757277908895 708545843879757744654044302530777598 Male 8595616048369598656677424462769380

35. SPSS Solution H 0 : µ1 = µ2 H 1: µ1  µ2 Fadwa Flemban35

36. Analyze  Compare means  Independent Samples T Test Fadwa Flemban36

37. Fadwa Flemban37 Press Define Groups

38. Fadwa Flemban38 Press Option

39. Output : Fadwa Flemban39 0.771 > 0.05 Equal Variance Assumed 0.771 > 0.05 Equal Variance Assumed Inside the Box: T-Test for Equality of means t=.045, df =(33+35)-2 =68-2= 66 Sig.=0.964> α=0.05 Don’t reject H Confidence Interval of the difference (-9.949,10.412) Inside the Box: T-Test for Equality of means t=.045, df =(33+35)-2 =68-2= 66 Sig.=0.964> α=0.05 Don’t reject H Confidence Interval of the difference (-9.949,10.412)

40. What is learned: Test of Hypotheses : T-Test of two Independent Samples. Estimate : Confidence Interval of the difference between means. Mean difference : µ1-µ2 = 64.69 – 64.45 = 0.24 ( from Group Statistics Table ) Or mean difference = 0.23 1 ( from Independent samples test Table ) Fadwa Flemban40

41. Make a decision : Fadwa Flemban41 1) From P-value (Sig.) P-value > α Don’t reject H P-value < α Reject H 2) From Confidence Interval 0 inside the Interval Don’t reject H 0 outside the Interval Reject H

42. Tests of Differences between Means, dependent Samples Example : We have the data of 20 students in two courses (Arabic & English): Examine if there is difference between the mean of students’ marks in two courses, at level of significant 0.10? Fadwa Flemban42 10987654321Num 60 8090804085508085Arabic 3040849590506550 80English 20191817161514131211Num 85805560707590858087Arabic 75907040607570657580English

43. SPSS Solution H 0 : µ1 = µ2 H 1: µ1  µ2 Fadwa Flemban43

44. Fadwa Flemban44 Analyze  Compare means  Paired Samples T Test

45. Fadwa Flemban45 Press Option

46. Output: Fadwa Flemban46 t= 2.309, df=20-1=19 Sig.=0.032 < α=0.10 Reject H Confidence Interval of the difference (1.796,12.504) Note: the interval doesn’t contain ZERO t= 2.309, df=20-1=19 Sig.=0.032 < α=0.10 Reject H Confidence Interval of the difference (1.796,12.504) Note: the interval doesn’t contain ZERO

47. اخطاء شائعة استخدام اختبارات t : لبيانات صغيرة غير مسحوبة من مجتمع طبيعي Fadwa Flemban47

48. Statistical Humor A boy asked his statistician father, "Why is my body not well proportioned just like my brother's?" His father's response, "Because, when your mother had your pregnancy, its distribution was skewed!!" Fadwa Flemban48