Mathematics 10 — Exercise 11.2 · Q8 (i–ii)

Full question

Compare machine outputs for average performance and consistency. A: [98, 100, 102, 101, 99]. B: [95, 100, 105, 90, 110].

Step-by-step solution

1. Calculate the statistics for A independently.

2. Multiply the value (or class midpoint) by its frequency.

   $$1\times 98=98$$

3. Square that value.

   $${98}^2=9604$$

4. Multiply the square by its frequency.

   $$1\times 9604=9604$$

5. Multiply the value (or class midpoint) by its frequency.

   $$1\times 100=100$$

6. Square that value.

   $${100}^2=10000$$

7. Multiply the square by its frequency.

   $$1\times 10000=10000$$

8. Multiply the value (or class midpoint) by its frequency.

   $$1\times 102=102$$

9. Square that value.

   $${102}^2=10404$$

10. Multiply the square by its frequency.

    $$1\times 10404=10404$$

11. Multiply the value (or class midpoint) by its frequency.

    $$1\times 101=101$$

12. Square that value.

    $${101}^2=10201$$

13. Multiply the square by its frequency.

    $$1\times 10201=10201$$

14. Multiply the value (or class midpoint) by its frequency.

    $$1\times 99=99$$

15. Square that value.

    $${99}^2=9801$$

16. Multiply the square by its frequency.

    $$1\times 9801=9801$$

17. Add the frequencies to find n.

    $$1+1+1+1+1=5$$

18. Add the f×x column.

    $$98+100+102+101+99=500$$

19. Add the f×x² column.

    $$9604+10000+10404+10201+9801=50010$$

20. Mean = Σfx/Σf. Divide the first total by n.

    $$\frac{500}{5}$$

21. Calculate this operation before combining the remaining terms.

    $$500\times 1=500$$

22. Calculate this operation before combining the remaining terms.

    $$1\times 5=5$$

23. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{500\times 1}{1\times 5}$$

24. Calculate these two products separately.

    $$\frac{500}{5}$$

25. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=5$$

26. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{5\times 1}{1\times 5}$$

27. Calculate these two products separately.

    $$\frac{5}{5}$$

28. Show the two divisions explicitly.

    $$\frac{\frac{5}{5}}{\frac{5}{5}}$$

29. Write the reduced fraction.

    $$\frac{1}{1}$$

30. Calculate this operation before combining the remaining terms.

    $$5\times \frac{1}{5}=1$$

31. Calculate this operation before combining the remaining terms.

    $$\frac{1}{1}=1$$

32. Calculate this operation before combining the remaining terms.

    $$100\times 1=100$$

33. Show the two divisions explicitly.

    $$\frac{\frac{500}{5}}{\frac{5}{5}}$$

34. Evaluate the numerator division.

    $$\frac{500}{5}=100$$

35. Evaluate the denominator division.

    $$\frac{5}{5}=1$$

36. Write the reduced fraction.

    $$100\times \frac{1}{1}$$

37. Calculate this operation before combining the remaining terms.

    $$500\times \frac{1}{5}=100$$

38. Population variance = Σfx²/n − (mean)². Keep the exact mean before rounding.

    $$\frac{50010}{5}-{100}^2$$

39. Calculate this operation before combining the remaining terms.

    $$50010\times 1=50010$$

40. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{50010\times 1}{1\times 5}$$

41. Calculate these two products separately.

    $$\frac{50010}{5}$$

42. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=5$$

43. Calculate this operation before combining the remaining terms.

    $$10002\times 1=10002$$

44. Show the two divisions explicitly.

    $$\frac{\frac{50010}{5}}{\frac{5}{5}}$$

45. Evaluate the numerator division.

    $$\frac{50010}{5}=10002$$

46. Evaluate the denominator division.

    $$\frac{5}{5}=1$$

47. Write the reduced fraction.

    $$10002\times \frac{1}{1}$$

48. Calculate this operation before combining the remaining terms.

    $$50010\times \frac{1}{5}=10002$$

49. Calculate this operation before combining the remaining terms.

    $${100}^2=10000$$

50. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 10000=-10000$$

51. Calculate this operation before combining the remaining terms.

    $$10002-10000=2$$

52. Standard deviation is the POSITIVE square root of the variance.

    $$\sqrt{2}$$

53. Round the standard deviation only now.

    $$1.41421$$

54. Group A: mean=100.00000, standard deviation=1.41421.

55. Calculate the statistics for B independently.

56. Multiply the value (or class midpoint) by its frequency.

    $$1\times 95=95$$

57. Square that value.

    $${95}^2=9025$$

58. Multiply the square by its frequency.

    $$1\times 9025=9025$$

59. Multiply the value (or class midpoint) by its frequency.

    $$1\times 100=100$$

60. Square that value.

    $${100}^2=10000$$

61. Multiply the square by its frequency.

    $$1\times 10000=10000$$

62. Multiply the value (or class midpoint) by its frequency.

    $$1\times 105=105$$

63. Square that value.

    $${105}^2=11025$$

64. Multiply the square by its frequency.

    $$1\times 11025=11025$$

65. Multiply the value (or class midpoint) by its frequency.

    $$1\times 90=90$$

66. Square that value.

    $${90}^2=8100$$

67. Multiply the square by its frequency.

    $$1\times 8100=8100$$

68. Multiply the value (or class midpoint) by its frequency.

    $$1\times 110=110$$

69. Square that value.

    $${110}^2=12100$$

70. Multiply the square by its frequency.

    $$1\times 12100=12100$$

71. Add the frequencies to find n.

    $$1+1+1+1+1=5$$

72. Add the f×x column.

    $$95+100+105+90+110=500$$

73. Add the f×x² column.

    $$9025+10000+11025+8100+12100=50250$$

74. Mean = Σfx/Σf. Divide the first total by n.

    $$\frac{500}{5}$$

75. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{500\times 1}{1\times 5}$$

76. Calculate these two products separately.

    $$\frac{500}{5}$$

77. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=5$$

78. Show the two divisions explicitly.

    $$\frac{\frac{500}{5}}{\frac{5}{5}}$$

79. Evaluate the numerator division.

    $$\frac{500}{5}=100$$

80. Evaluate the denominator division.

    $$\frac{5}{5}=1$$

81. Write the reduced fraction.

    $$100\times \frac{1}{1}$$

82. Calculate this operation before combining the remaining terms.

    $$500\times \frac{1}{5}=100$$

83. Population variance = Σfx²/n − (mean)². Keep the exact mean before rounding.

    $$\frac{50250}{5}-{100}^2$$

84. Calculate this operation before combining the remaining terms.

    $$50250\times 1=50250$$

85. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{50250\times 1}{1\times 5}$$

86. Calculate these two products separately.

    $$\frac{50250}{5}$$

87. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=5$$

88. Calculate this operation before combining the remaining terms.

    $$10050\times 1=10050$$

89. Show the two divisions explicitly.

    $$\frac{\frac{50250}{5}}{\frac{5}{5}}$$

90. Evaluate the numerator division.

    $$\frac{50250}{5}=10050$$

91. Evaluate the denominator division.

    $$\frac{5}{5}=1$$

92. Write the reduced fraction.

    $$10050\times \frac{1}{1}$$

93. Calculate this operation before combining the remaining terms.

    $$50250\times \frac{1}{5}=10050$$

94. Calculate this operation before combining the remaining terms.

    $${100}^2=10000$$

95. Calculate this operation before combining the remaining terms.

    $$\left(-1\right)\times 10000=-10000$$

96. Calculate this operation before combining the remaining terms.

    $$10050-10000=50$$

97. Extract a perfect-square factor from under the root.

    $$50=25\times 2$$

98. Use √(ab)=√a×√b for nonnegative a,b; take the root of the perfect square.

    $$\sqrt{50}=5\times \sqrt{2}$$

99. Standard deviation is the POSITIVE square root of the variance.

    $$\sqrt{50}$$

100. Round the standard deviation only now.

     $$7.07107$$

101. Group B: mean=100.00000, standard deviation=7.07107.

102. Compare means for average performance or expenditure; compare standard deviations for consistency. A smaller standard deviation means less variation.

Answer

Higher average: equal. More consistent: A.