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.
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You're on a roll — here's the solution to problem 4. Problem 4: The mean height of three groups of students consisting of 20, 16 and 14 students are 1.67 m, 1.50 m and 1.40 m, respectively. Find the mean height of all the students. Calculate the mean, correct to two significant figures. To find the mean height of all students, we need to calculate the weighted mean. The formula for the weighted mean is: X_total = N_1 X_1 + N_2 X_2 + N_3 X_3N_1 + N_2 + N_3 where N_i is the number of students in group i and X_i is the mean height of group i. Step 1: Identify the given values for each group. For Group 1: N_1 = 20 students, X_1 = 1.67 m For Group 2: N_2 = 16 students, X_2 = 1.50 m For Group 3: N_3 = 14 students, X_3 = 1.40 m Step 2: Calculate the total number of students. Total students = N_1 + N_2 + N_3 = 20 + 16 + 14 = 50 Step 3: Calculate the sum of heights for each group and then the total sum of heights. Sum of heights for Group 1 = 20 × 1.67 m = 33.4 m Sum of heights for Group 2 = 16 × 1.50 m = 24.0 m Sum of heights for Group 3 = 14 × 1.40 m = 19.6 m Total sum of heights = 33.4 m + 24.0 m + 19.6 m = 77.0 m Step 4: Calculate the mean height of all the students. X_total = 77.0 m50 X_total = 1.54 m Step 5: Round the mean height to two significant figures as requested. The first two significant figures are 1 and 5. The next digit is 4, which is less than 5, so we round down. X_total ≈ 1.5 m The mean height of all the students, correct to two significant figures, is 1.5 m. What's next?