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 Plex — been a while! Let's pick up where you left off. 1. Calculate the Pearson product-moment correlation coefficient (r) Step 1: Create a table to calculate X, Y, XY, X^2, and Y^2. Given data: English (X): 1, 2, 3, 4, 5 Mathematics (Y): 2, 4, 5, 4, 5 Number of students, n = 5. |c|c|c|c|c| X & Y & XY & X^2 & Y^2 \\ 1 & 2 & 2 & 1 & 4 \\ 2 & 4 & 8 & 4 & 16 \\ 3 & 5 & 15 & 9 & 25 \\ 4 & 4 & 16 & 16 & 16 \\ 5 & 5 & 25 & 25 & 25 \\ X = 15 & Y = 20 & XY = 66 & X^2 = 55 & Y^2 = 86 \\ Step 2: Apply the Pearson correlation coefficient formula. The formula for Pearson product-moment correlation coefficient (r) is: r = (n XY - ( X)( Y))/(sqrt([n X^2 - ( X)^2][n Y^2 - ( Y)^2])) Substitute the calculated values: r = (5(66) - (15)(20))/(sqrt([5(55) - (15)^2][5(86) - (20)^2])) Step 3: Simplify the expression. r = (330 - 300)/(sqrt([275 - 225][430 - 400])) r = (30)/(sqrt([50][30])) r = (30)/(sqrt(1500)) r = (30)/(38.7298) r ≈ 0.7746 The Pearson product-moment correlation coefficient is 0.7746. 2. Find the variance and standard deviation of the following set of data: 5, 8, 12, 15 Step 1: Calculate the mean (X) of the data. Given data: X = \5, 8, 12, 15\ Number of data points, n = 4. X = 5 + 8 + 12 + 15 = 40 X = ( X)/(n) = (40)/(4) = 10 Step 2: Calculate the sum of squared differences from the mean, (X - X)^2. |c|c|c| X & X - X & (X - X)^2 \\ 5 & 5 - 10 = -5 & 25 \\ 8 & 8 - 10 = -2 & 4 \\ 12 & 12 - 10 = 2 & 4 \\ 15 & 15 - 10 = 5 & 25 \\ (X - X)^2 = 58 \\ Step 3: Calculate the variance (^2). Assuming this is a population, the variance formula is: ^2 = (X - X)^2n ^2 = (58)/(4) = 14.5 Step 4: Calculate the standard deviation (). The standard deviation is the square root of the variance: = sqrt(^2) = sqrt(14.5) ≈ 3.8079 The variance is 14.5 and the standard deviation is 3.8079. 4. Using Spearman's rank, obtain the correlation coefficient of the data set below. Step 1: Rank the X (Theory) and Y (Practical) values separately. Given data: Theory (X): 8, 3, 9, 2, 7, 10, 4, 6, 1, 5 Practical (Y): 9, 5, 10, 1, 8, 7, 3, 4, 2, 6 Number of data pairs, n = 10. Step 2: Create a table to calculate the ranks, differences in ranks (d), and squared differences (d^2). |c|c|c|c|c|c| X & Y & Rank X (Rx) & Rank Y (Ry) & d = Rx - Ry & d^2 \\ 8 & 9 & 8 & 9 & -1 & 1 \\ 3 & 5 & 3 & 5 & -2 & 4 \\ 9 & 10 & 9 & 10 & -1 & 1 \\ 2 & 1 & 2 & 1 & 1 & 1 \\ 7 & 8 & 7 & 8 & -1 & 1 \\ 10 & 7 & 10 & 7 & 3 & 9 \\ 4 & 3 & 4 & 3 & 1 & 1 \\ 6 & 4 & 6 & 4 & 2 & 4 \\ 1 & 2 & 1 & 2 & -1 & 1 \\ 5 & 6 & 5 & 6 & -1 & 1 \\ d^2 = 24 \\ Step 3: Apply Spearman's rank correlation coefficient formula. The formula for Spearman's rank correlation coefficient (r_s) is: r_s = 1 - (6 d^2)/(n(n^2 - 1)) Substitute n = 10 and d^2 = 24: r_s = 1 - (6(24))/(10(10^2 - 1)) r_s = 1 - (144)/(10(100 - 1)) r_s = 1 - (144)/(10(99)) r_s = 1 - (144)/(990) r_s = 1 - 0.145454... r_s ≈ 0.8545 The Spearman's rank correlation coefficient is 0.8545. Drop the next question! 📸