a) Find the correlation coefficient between X and Y. X = [45, 50, 56, 62, 65, 70], Y = [65, 70, 75, 80, 90, 100]

Here is the solution for Question 16. Question 16: a) Find the correlation coefficient between X and Y. Step 1: List the given data for X (supply) and Y (price). X = [45, 50, 56, 62, 65, 70] Y = [65, 70, 75, 80, 90, 100] The number of data points is n=6. Step 2: Calculate the sums needed for the correlation coefficient formula. • X = 45+50+56+62+65+70 = 348 • Y = 65+70+75+80+90+100 = 480 • XY = (45 × 65) + (50 × 70) + (56 × 75) + (62 × 80) + (65 × 90) + (70 × 100) XY = 2925 + 3500 + 4200 + 4960 + 5850 + 7000 = 28435 • X^2 = 45^2 + 50^2 + 56^2 + 62^2 + 65^2 + 70^2 X^2 = 2025 + 2500 + 3136 + 3844 + 4225 + 4900 = 20630 • Y^2 = 65^2 + 70^2 + 75^2 + 80^2 + 90^2 + 100^2 Y^2 = 4225 + 4900 + 5625 + 6400 + 8100 + 10000 = 39250 Step 3: Apply the Pearson correlation coefficient formula. The formula is: r = (n XY - ( X)( Y))/(sqrt([n X^2 - ( X)^2][n Y^2 - ( Y)^2])) Substitute the calculated values: r = (6(28435) - (348)(480))/(sqrt([6(20630) - (348)^2][6(39250) - (480)^2])) r = (170610 - 167040)/(sqrt([123780 - 121104][235500 - 230400])) r = (3570)/(sqrt([2676][5100])) r = (3570)/(sqrt(13647600)) r = (3570)/(3694.2656) r ≈ 0.9664 The correlation coefficient between X and Y is 0.9664. b) Calculate the supply when the price of commodity is Rs. 150. Step 1: Determine the regression line equation of X on Y (X = a + bY). First, calculate the slope b: b = (n XY - ( X)( Y))/(n Y^2 - ( Y)^2) Using values from part (a): b = (3570)/(5100) = 0.7 Next, calculate the means X and Y: X = ( X)/(n) = (348)/(6) = 58 Y = ( Y)/(n) = (480)/(6) = 80 Now, calculate the y-intercept a: a = X - bY a = 58 - (0.7)(80) a = 58 - 56 a = 2 The regression equation of X on Y is X = 2 + 0.7Y. Step 2: Calculate the supply (X) when the price (Y) is Rs. 150. Substitute Y = 150 into the regression equation: X = 2 + 0.7(150) X = 2 + 105 X = 107 The supply when the price of the commodity is Rs. 150 is 107. 3 done, 2 left today. You're making progress.