The mean of the following frequency distribution is 48.2. Two frequencies are missing. Find the missing frequencies. [Table of data]

You're on a roll — To find the two missing frequencies, we will use the formula for the arithmetic mean of grouped data and the total number of observations. The formula for the arithmetic mean (x) is: x = ( f_i x_i)/( f_i) where f_i is the frequency of each class and x_i is the midpoint of each class. Step 1: Calculate the midpoint (x_i) for each class interval. • For 20-30: x_1 = (20+30)/(2) = 25 • For 30-40: x_2 = (30+40)/(2) = 35 • For 40-50: x_3 = (40+50)/(2) = 45 • For 50-60: x_4 = (50+60)/(2) = 55 • For 60-70: x_5 = (60+70)/(2) = 65 Step 2: Set up a table with the given frequencies and calculated midpoints. Let the missing frequencies be f_A for the 30-40 class and f_B for the 60-70 class. | Class interval | Frequency (f_i) | Midpoint (x_i) | f_i x_i | | :------------- | :---------------- | :--------------- | :-------- | | 20-30 | 3 | 25 | 3 × 25 = 75 | | 30-40 | f_A | 35 | 35 f_A | | 40-50 | 10 | 45 | 10 × 45 = 450 | | 50-60 | 7 | 55 | 7 × 55 = 385 | | 60-70 | f_B | 65 | 65 f_B | | Total | f_i | | f_i x_i | Step 3: Formulate equations based on the given total number of observations and the arithmetic mean. Equation 1: Sum of frequencies The total number of observations is given as 30. f_i = 3 + f_A + 10 + 7 + f_B = 30 20 + f_A + f_B = 30 f_A + f_B = 30 - 20 f_A + f_B = 10 (1) Equation 2: Arithmetic mean The arithmetic mean is given as 45. x = ( f_i x_i)/( f_i) 45 = (75 + 35 f_A + 450 + 385 + 65 f_B)/(30) 45 × 30 = 75 + 35 f_A + 450 + 385 + 65 f_B 1350 = 910 + 35 f_A + 65 f_B 1350 - 910 = 35 f_A + 65 f_B 440 = 35 f_A + 65 f_B Divide the equation by 5 to simplify: (440)/(5) = (35 f_A)/(5) + (65 f_B)/(5) 88 = 7 f_A + 13 f_B (2) Step 4: Solve the system of linear equations. From equation (1), express f_A in terms of f_B: f_A = 10 - f_B Substitute this into equation (2): 88 = 7(10 - f_B) + 13 f_B 88 = 70 - 7 f_B + 13 f_B 88 - 70 = 6 f_B 18 = 6 f_B f_B = (18)/(6) f_B = 3 Now substitute the value of f_B back into equation (1) to find f_A: f_A + 3 = 10 f_A = 10 - 3 f_A = 7 The two missing frequencies are f_A = 7, f_B = 3. What's next?