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
Okay, Shasha, let's tackle this grouped data problem. There's an inconsistency in the first row of the table provided (the given Midpoint (M) and f(M) don't match the frequency (f) or the class interval). I will proceed by calculating the Midpoint (M) for each class interval using the standard formula M = lower limit + upper limit2 and then calculate f(M) accordingly. I will use these consistent values to complete the table and solve the subsequent questions. C2. Complete the table below: Step 1: Calculate the Midpoint (M) for each class interval. • For 37-38: M = (37+38)/(2) = 37.5 • For 39-40: M = (39+40)/(2) = 39.5 • For 41-42: M = (41+42)/(2) = 41.5 • For 43-44: M = (43+44)/(2) = 43.5 • For 45-46: M = (45+46)/(2) = 45.5 • For 47-48: M = (47+48)/(2) = 47.5 • For 49-50: M = (49+50)/(2) = 49.5 Step 2: Calculate f(M) for each row by multiplying Frequency (f) by Midpoint (M). • For 37-38: f(M) = 3 × 37.5 = 112.5 • For 39-40: f(M) = 3 × 39.5 = 118.5 • For 41-42: f(M) = 3 × 41.5 = 124.5 • For 43-44: f(M) = 4 × 43.5 = 174.0 • For 45-46: f(M) = 8 × 45.5 = 364.0 • For 47-48: f(M) = 4 × 47.5 = 190.0 • For 49-50: f(M) = 3 × 49.5 = 148.5 Step 3: Calculate the total frequency (N) and the sum of f(M) ( fM). • N = 3+3+3+4+8+4+3 = 28 • fM = 112.5 + 118.5 + 124.5 + 174.0 + 364.0 + 190.0 + 148.5 = 1232.0 Here is the completed table: | Class Interval | Frequency (f) | Midpoint (M) | f(M) | | :------------- | :------------ | :----------- | :--- | | 37-38 | 3 | 37.5 | 112.5 | | 39-40 | 3 | 39.5 | 118.5 | | 41-42 | 3 | 41.5 | 124.5 | | 43-44 | 4 | 43.5 | 174.0 | | 45-46 | 8 | 45.5 | 364.0 | | 47-48 | 4 | 47.5 | 190.0 | | 49-50 | 3 | 49.5 | 148.5 | | N= | 28 | | fM= 1232.0 | Now, using the information from the completed table: a) Mean of the grouped data Step 1: Use the formula for the mean of grouped data. x = ( fM)/(N) Step 2: Substitute the calculated values. x = (1232.0)/(28) Step 3: Calculate the mean. x = 44 The mean of the grouped data is 44. b) Median of the grouped data Step 1: Find the median position. (N)/(2) = (28)/(2) = 14 The 14th value falls in the class interval 45-46. This is the median class. Step 2: Identify the components for the median formula. • Lower class boundary (L) of the median class (45-46): 44.5 • Cumulative frequency before the median class (CF_b): 3+3+3+4 = 13 (for class 43-44) • Frequency of the median class (f_m): 8 • Class width (w): 38-37+1 = 2 (or 39-37=2) Step 3: Apply the median formula for grouped data. Median = L + ((N)/(2) - CF_bf_m) × w Median = 44.5 + ((14 - 13)/(8)) × 2 Median = 44.5 + ((1)/(8)) × 2 Median = 44.5 + 0.125 × 2 Median = 44.5 + 0.25 Median = 44.75 The median of the grouped data is 44.75. c) Mode of the grouped data Step 1: Identify the modal class (the class with the highest frequency). The highest frequency is 8, which corresponds to the class interval 45-46. This is the modal class. Step 2: Identify the components for the mode formula. • Lower class boundary (L) of the modal class (45-46): 44.5 • Frequency of the modal class (f_m): 8 • Frequency of the class before the modal class (f_1): 4 (for class 43-44) • Frequency of the class after the modal class (f_2): 4 (for class 47-48) • Class width (w): 2 Step 3: Apply the mode formula for grouped data. Mode = L + ((f_m - f_1)/((f_m - f_1) + (f_m - f_2))) × w Mode = 44.5 + ((8 - 4)/((8 - 4) + (8 - 4))) × 2 Mode = 44.5 + ((4)/(4 + 4)) × 2 Mode = 44.5 + ((4)/(8)) × 2 Mode = 44.5 + 0.5 × 2 Mode = 44.5 + 1 Mode = 45.5 The mode of the grouped data is 45.5. d) Range of the grouped data Step 1: Identify the upper class boundary of the highest class and the lower class boundary of the lowest class. • Highest class interval: 49-50. Upper class boundary = 50.5 • Lowest class interval: 37-38. Lower class boundary = 36.5 Step 2: Calculate the range. Range = Upper boundary of highest class - Lower boundary of lowest class Range = 50.5 - 36.5 Range = 14 The range of the grouped data is 14. Last free one today — make it count tomorrow, or type /upgrade for unlimited.