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
a) To calculate the product-moment correlation coefficient r, we first need to find the sums: X, Y, X^2, Y^2, and XY. There are n=10 competitors. Step 1: Calculate the sums. X = 10+12+16+13+17+11+18+15+19+14 = 145 Y = 18+15+10+16+19+14+13+11+17+12 = 145 X^2 = 10^2+12^2+16^2+13^2+17^2+11^2+18^2+15^2+19^2+14^2 = 2185 Y^2 = 18^2+15^2+10^2+16^2+19^2+14^2+13^2+11^2+17^2+12^2 = 2185 XY = (10 × 18) + (12 × 15) + (16 × 10) + (13 × 16) + (17 × 19) + (11 × 14) + (18 × 13) + (15 × 11) + (19 × 17) + (14 × 12) = 2095 Step 2: Apply the formula for the product-moment correlation coefficient. r = (n XY - ( X)( Y))/(sqrt([n X^2 - ( X)^2][n Y^2 - ( Y)^2])) Substitute the calculated values: r = (10(2095) - (145)(145))/(sqrt([10(2185) - (145)^2][10(2185) - (145)^2])) r = (20950 - 21025)/(sqrt([21850 - 21025][21850 - 21025])) r = (-75)/(sqrt([825][825])) r = (-75)/(825) r = -0.090909... Rounding to 2 decimal places: r = -0.09 The product-moment correlation coefficient is -0.09. b) To determine the least square regression line of the marks in fashion parade (Y) on the marks in dancing (X), we use the equation Y = a + bX. Step 1: Calculate the slope b. b = (n XY - ( X)( Y))/(n X^2 - ( X)^2) Using the values from part a): b = (-75)/(825) b = -(1)/(11) Step 2: Calculate the y-intercept a. First, find the means X and Y: X = ( X)/(n) = (145)/(10) = 14.5 Y = ( Y)/(n) = (145)/(10) = 14.5 Now, use the formula a = Y - bX: a = 14.5 - (-(1)/(11))(14.5)