This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Welcome back — missed you this week. 1. a) Range is a measure of dispersion in statistics that represents the difference between the highest and lowest values in a dataset. It provides a quick, but often crude, indication of the spread of data. While easy to calculate, it is highly sensitive to outliers and does not provide information about the distribution of values between the extremes. 1. b) The scores are 2, 5, 6, 4, 5. Step 1: Calculate the mean (x). x = (2 + 5 + 6 + 4 + 5)/(5) = (22)/(5) = 4.4 Step 2: Calculate the absolute deviations from the mean (|x_i - x|). |2 - 4.4| = 2.4 |5 - 4.4| = 0.6 |6 - 4.4| = 1.6 |4 - 4.4| = 0.4 |5 - 4.4| = 0.6 Step 3: Calculate the sum of the absolute deviations. |x_i - x| = 2.4 + 0.6 + 1.6 + 0.4 + 0.6 = 5.6 Step 4: Calculate the mean deviation. Mean Deviation = |x_i - x|n = (5.6)/(5) = 1.12 The mean deviation is 1.12. 1. c) Given data: X: 4, 8, 9, 6, 2 Y: 3, 7, 6, 9, 2 Number of observations, n = 5. Step 1: Create a table to calculate necessary sums. | X | Y | XY | X^2 | Y^2 | |-----|-----|------|-------|-------| | 4 | 3 | 12 | 16 | 9 | | 8 | 7 | 56 | 64 | 49 | | 9 | 6 | 54 | 81 | 36 | | 6 | 9 | 54 | 36 | 81 | | 2 | 2 | 4 | 4 | 4 | |-----|-----|------|-------|-------| | X = 29 | Y = 27 | XY = 180 | X^2 = 201 | Y^2 = 179 | Step 2: Apply Pearson's correlation coefficient formula. r = (n XY - ( X)( Y))/(sqrt([n X^2 - ( X)^2][n Y^2 - ( Y)^2])) r = (5(180) - (29)(27))/(sqrt([5(201) - (29)^2][5(179) - (27)^2])) r = (900 - 783)/(sqrt([1005 - 841][895 - 729])) r = (117)/(sqrt([164][166])) r = (117)/(sqrt(27224)) r = (117)/(164.996969...) r ≈ 0.7091 The coefficient of correlation is 0.7091. 1. d) When a fair die is tossed twice, the total number of possible outcomes is 6 × 6 = 36. The outcomes that result in a total of 5 are: (1, 4), (2, 3), (3, 2), (4, 1) There are 4 favorable outcomes. The probability of obtaining a total of 5 is: P(total of 5) = Number of favorable outcomesTotal number of outcomes = (4)/(36) = (1)/(9) The probability is *(1)/(9) or 0.1111. 2. Given data: X: 1, 2, 4, 5, 8, 10, 12, 14, 16 Y: 9, 8, 10, 12, 14, 15, 16, 18, 15 Number of observations, n = 9. Step 1: Create a table to calculate necessary sums. | X | Y | XY | X^2 | |-----|-----|------|-------| | 1 | 9 | 9 | 1 | | 2 | 8 | 16 | 4 | | 4 | 10 | 40 | 16 | | 5 | 12 | 60 | 25 | | 8 | 14 | 112 | 64 | | 10 | 15 | 150 | 100 | | 12 | 16 | 192 | 144 | | 14 | 18 | 252 | 196 | | 16 | 15 | 240 | 256 | |-----|-----|------|-------| | X = 72 | Y = 117 | XY = 1071 | X^2 = 806 | Step 2: Calculate the slope (b) of the regression line Y = a + bX. b = (n XY - ( X)( Y))/(n X^2 - ( X)^2) b = (9(1071) - (72)(117))/(9(806) - (72)^2) b = (9639 - 8424)/(7254 - 5184) b = (1215)/(2070) ≈ 0.5869565 Step 3: Calculate the means X and Y. X = ( X)/(n) = (72)/(9) = 8 Y = ( Y)/(n) = (117)/(9) = 13 Step 4: Calculate the Y-intercept (a). a = Y - bX a = 13 - (0.5869565)(8) a = 13 - 4.695652 a = 8.304348 Step 5: Write the regression equation. Y = 8.3043 + 0.5870X The line of regression of Y on X is Y = 8.3043 + 0.5870X. 3. In a single toss of a die, the sample space is S = \1, 2, 3, 4, 5, 6\. The total number of outcomes is 6. Step 1: Identify even numbers. Even numbers are E = \2, 4, 6\. Step 2: Identify composite numbers. A composite number is a positive integer that has at least one divisor other than 1 and itself. From the sample space: • 1 is neither prime nor composite. • 2 is prime. • 3 is prime. • 4 is composite (divisors: 1, 2, 4). • 5 is prime. • 6 is composite (divisors: 1, 2, 3, 6). So, composite numbers are C = \4, 6\. Step 3: Identify numbers that are even OR composite (E C). E C = \2, 4