The table shows the results from the survey of a number of people. Find the value of a and b. Mean = 9.28

Here's the solution to the problem: Part (a) a) (i) Determine the value of a and b. Step 1: Calculate the total number of inductors. The total number of inductors is given as 50. 2 + a + 12 + 15 + b + 3 = 50 32 + a + b = 50 a + b = 50 - 32 a + b = 18 (Equation 1) Step 2: Calculate the midpoints (x_i) for each class interval. 3 - 5: (3+5)/2 = 4 5 - 7: (5+7)/2 = 6 7 - 9: (7+9)/2 = 8 9 - 11: (9+11)/2 = 10 11 - 13: (11+13)/2 = 12 13 - 15: (13+15)/2 = 14 Step 3: Use the mean formula to set up a second equation. The mean (x) for grouped data is given by x = ( f_i x_i)/( f_i). We are given x = 9.28 and f_i = 50. So, f_i x_i = 9.28 × 50 = 464. Now, calculate f_i x_i: (2 × 4) + (a × 6) + (12 × 8) + (15 × 10) + (b × 12) + (3 × 14) = 464 8 + 6a + 96 + 150 + 12b + 42 = 464 296 + 6a + 12b = 464 6a + 12b = 464 - 296 6a + 12b = 168 Divide the equation by 6: a + 2b = 28 (Equation 2) Step 4: Solve the system of equations for a and b. Equation 1: a + b = 18 Equation 2: a + 2b = 28 Subtract Equation 1 from Equation 2: (a + 2b) - (a + b) = 28 - 18 b = 10 Substitute b=10 into Equation 1: a + 10 = 18 a = 18 - 10 a = 8 The values are: a=8, b=10 a) (ii) Determine the mode. Step 1: Update the frequency table with the values of a and b. Inductance (mH) | Number of inductors (f_i) ------------------|---------------------------- 3 - 5 | 2 5 - 7 | 8 7 - 9 | 12 9 - 11 | 15 11 - 13 | 10 13 - 15 | 3 Step 2: Identify the modal class. The modal class is the class with the highest frequency. In this case, the highest frequency is 15, which corresponds to the class interval 9 - 11. Step 3: Apply the formula for the mode of grouped data. The formula for the mode is: Mode = L + ( (f_m - f_1)/(2f_m - f_1 - f_2) ) × w Where: L = 9 (lower boundary of the modal class) f_m = 15 (frequency of the modal class) f_1 = 12 (frequency of the class preceding the modal