Mathematics in modern world subject normal distribution

For the gowww

1. Normal Distribution

2. NORMAL CURVE DISTRIBUTION Normal distribution , sometimes called the bell curve , is a  distribution  that occurs naturally in many situations.

3. Properties of Normal Distribution: Mean = median = mode The normal curve is a bell-shaped and symmetric about the mean The area to left of the y-axis is 50% and the area to right of the y-axis is 50%. The total area under the curve is equal to 1 or 100% The two end tail of the curve never touches the x-axis as it extends from the mean.

4. FINDING THE AREA UNDER THE NORMAL CURVE

5. Case #1: Finding the Area from 0 to z-scores  

6. Case #1: Finding the area from 0 to z-scores   Example #1: Find the area from 0 to 1.55  

7. Case #1: Finding the area from 0 to z-scores   Example #2: Find the area from 0 to 1.48  

8. Case #1: Finding the area from 0 to z-scores   Example #3: Find the area from 0 to 2.83

9. Case #1: Finding the area from 0 to z-scores   Example #4: Find the area from 0 to -0.28

10. Case #2: Finding the Area to the left/right of z-scores  

11. Case #2: Finding the area to the left/right of z-scores   Example #1: Find the area to the left of 1.55

12. Case #2: Finding the area to the left/right of z-scores   Example #2: Find the area to the left of -1.37

13. Note: Area to the left of (+)z-score Area to the right of ( - ) z-score Area to the left of (-) z-score Area to the right of ( + ) z-score add 0.5000 subtract 0.5000

14. Case #2: Finding the area to the left/right of z-scores   Example #3: Find the area to the right of 2.75

15. Case #2: Finding the area to the left/right of z-scores   Example #4: Find the area to the right of -0.82

16. Case #3: Finding the Area between two (2) z-scores  

17. Note: Area bet. + and + Area bet. – and - Area bet. + and - Area bet. – and + subtract the two areas add the two areas

18. Case #3: Finding the area between two z-scores   Example #1: Find the area between z= 2.53 to z = -0.57

19. Case #3: Finding the area between two z-scores   Example #2: Find the area between z= 1.75 to z= 0.10

20. Case #3: Finding the area between two z-scores   Example #3: Find the area between z= -0.45 to z= -1.65

21. Case #3: Finding the area between two z-scores   Example #4: Find the area between z= -0.35 to z= 1.30

22. Z-scores/ Standard Score

23. Z-scores It is a numerical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. Z-scores may be positive or negative, with a positive value indicating the score is above the mean and a negative score indicating it is below the mean.

24. Z-scores Formula: or Where : z – standard score x – score µ/ - mean σ/s- standard deviation  

25. Example 1: You take the SAT and score 1100. The mean score for the SAT is 1026 and the standard deviation is 37. How well did you score on the test compared to the average test taker?

26. Example 2: The average height of students in a class is 165 cm with a standard deviation of 8cm. If a student’s height is 180cm, what is their z-score?

27. Example 3: A basketball player’s free throw success rate is mean= 60%, with a standard deviation of 12%. The player scored exactly 60% in one game. What is the z-score?

28. Example 4: A basketball player’s free throw success rate is mean= 60%, with a standard deviation of 12%. The player scored exactly 60% in one game. What is the z-score?

29. Mathematics of Finance (Interest)

30. Basic Terms Interest – is the amount charged for using borrowed money. Principal amount – is the sum money that is originally borrowed from an individual or financial institution. It does not include interest.

31. Basic Terms Final Amount – Combined amount of principal amount and interest

32. It is the interest on the amount invested or borrowed at a given rate and for a given time. Formula: I = Prt F = P + I Simple Interest Where: P – Principal amount F – final value/amount r – rate (%) t – time (in years) I - Interest

33. If rate is unknown If time is unknown   Derived Formula If Principal amount is unknown  

34. To buy a computer, Raquel borrowed ₱ 43,000 at 9% interest for 4 years. How much money did she have to pay back? Problem #1: Given: P = ₱43,000 r = 9% or 0.09 t = 4 years I = Prt I = (43,000)(0.09)(4) I = ₱15,480 F = 43000 + 15480 F = ₱58,480

35. April borrowed ₱ 2,250 from her friend in order to pay her parcel. Her friend told her that she will charge it with 15% interest. How much money interest will April pay after 6 months? Problem #2: Given: P = ₱2,250 r = 15% or 0.15 t = 6 mos. or or or 0.50 I = Prt I = (2250)(0.15)( ) I = ₱168.75  

36. Find the amount on ₱9,700 to be paid for 5 years and 3 months at 6% simple interest. Problem #3: Given: P = ₱9,700 r = 6% or 0.06 t = 5 yrs and 3 months or or 5.25 I = Prt I = (9700)(0.06)(5.25) I = ₱3,055.5  

37. Jessica invests $3,000 in a credit union at an interest rate of 3.9%. She leaves the money there for 5 years. What is her balance after that time? Problem #4: Given: P = $3,000 r = 3.9% or 0.039 t = 5 years I = Prt I = (3000)(0.039)(5) I = $585 F = 3000 + 585 = $ 3585

38. Mrs. Payne has ₱ 20,000 to invest. She wants to earn ₱ 10,000 in interest. She is considering a savings and loans bank that is offering her 5% interest per year. For how long will she have to leave her money in the bank in order to reach her goal of ₱ 10,000? Problem #5: Given: P = ₱20,000 r = 5% or 0.05 t = ? I = ₱10,000  

39. Mr. Zeus has ₱45,450 to invest. He wants to earn ₱12,125 in interest. He is considering a savings loan bank that offers 13.5% interest /year. For how long will he have to leave his money in the bank to reach his goal of ₱12,125? Problem #7: Given: P = ₱45,450 r = 13.5% or 0.135 t = ? I = ₱12125  

40. A nurse put ₱ 22,000 in the bank 15 years ago. She has earned ₱ 21,450 in interest—nearly as much as her initial investment. What was the interest rate that the bank was paying her? Problem #6: Given: P = ₱22,000 r = ? t = 15 I = ₱21,450  

41. Modular Arithmetic

42. Definition: Modular Arithmetic – is a system of arithmetic for integers, where “wrap around” upon reaching a certain value, the modulus (plural moduli). It is also known as “ taking remainder”

43. Dividing numbers: Remainder Quotient Dividend Divisor

44. Examples: 5 mod 3 8 mod 4 11 mod 3 9 mod 7 8 mod 5 10 mod 3 14 mod 5 100 mod 20 89 mod 12 Answers: = 2 = 0 = 2 = 2 = 3 = 1 = 4 = 0 = 5

45. Operations on Modular Arithmetic

46. Examples: (23 + 38) mod 12 (51 + 72) mod 3 (36 – 16) mod 6 (27 – 14) mod 5 (21 – 43) mod 7 (97 – 123) mod 8 (162 – 345) mod 13 (15 23) mod 11 (33 41) mod 17   Answers: = 1 = 0 = 2 = 3 = 6 = 6 = 12 = 4 = 10

47. Applications on Modular Arithmetic

48. Example (Time) Disregarding A.M. or P.M., if it is 5 o’clock now, what time will it be 57 hours from now? Solution: = (5 + 57) mod 12  

49. Example (Time) Disregarding A.M. or P.M., if it is 7 o’clock now, what time was it 51 hours ago? Solution: = (7 – 51) mod 12  

50. Example (Time) Disregarding A.M. or P.M., if it is 10 o’clock now, what time was it 83 hours ago? Solution: = (10 – 83) mod 12  

51. Example (Time) If it is 5 :00AM in the morning. What time will it be 42 hours from now?

52. Example (Time) If it is 3 :00AM in the morning. What time was it 66 hours ago?

53. Example (Time) If it is 9 :00PM in the evening. What time was it 49 hours ago?

54. Example (Time) If it is 6 :00PM in the evening. What time will it be 65 hours from now?

55. (Days of the week) Monday = 1 Tuesday = 2 Wednesday = 3 Thursday = 4 Friday = 5 Saturday = 6 Sunday = 7 or 0

56. Example (Days of the week) If today is Tuesday, what day of the week will it be 93 days from now? Solution: = (2 + 93) mod 7    

57. Example (Days of the week) If today is Monday, what day of the week was it 85 days ago? Solution: = (1 – 85) mod 7  

58. Example (Days of the week) If today is Wednesday, what day of the week was it 106 days ago? Solution: = (3 – 106) mod 7  

59. Example (Days of the week) If today is Saturday, what day of the week will it be 81 days from now? Solution: = (6 + 81) mod 7  

60. Exercise If today is Friday, what day of the week was it 32 days ago? Answer: Monday

61. Exercise What day will it be on June 16, 2035, if June 16, 2024 is Sunday?

62. Exercise What day will it be on October 12, 2029, if October 12, 2023 is Thursday?

63. Exercise What day will it be on April 9, 2028, if April 9, 2024 is Wednesday?

64. Exercise What day will it be on January 21 , 2032, if January 21, 2024 is Sunday?