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
Here's the solution for question 2 from Section B. 2 (i) Class of 100 students' marks a) Find the mean and median mark. Step 1: Calculate the sum of (mark × frequency) and total frequency for the mean. Let x be the mark and f be the frequency. f = 1+2+5+12+30+28+15+5+2 = 100 fx = (5 × 1) + (10 × 2) + (15 × 5) + (20 × 12) + (25 × 30) + (30 × 28) + (35 × 15) + (40 × 5) + (45 × 2) fx = 5 + 20 + 75 + 240 + 750 + 840 + 525 + 200 + 90 = 2745 Step 2: Calculate the mean. Mean = ( fx)/( f) = (2745)/(100) = 27.45 Step 3: Construct a cumulative frequency table to find the median. | Mark (x) | Frequency (f) | Cumulative Frequency (CF) | |:----------:|:---------------:|:-------------------------:| | 5 | 1 | 1 | | 10 | 2 | 3 | | 15 | 5 | 8 | | 20 | 12 | 20 | | 25 | 30 | 50 | | 30 | 28 | 78 | | 35 | 15 | 93 | | 40 | 5 | 98 | | 45 | 2 | 100 | Step 4: Determine the median. Since the total number of students is N=100 (an even number), the median is the average of the (N)/(2)-th and ((N)/(2)+1)-th values. The (100)/(2) = 50-th value. The ((100)/(2)+1) = 51-st value. From the cumulative frequency table: The 50th value is 25 (as CF reaches 50 at mark 25). The 51st value is 30 (as CF goes from 50 to 78 at mark 30). Median = 50th value + 51st value2 = (25 + 30)/(2) = (55)/(2) = 27.5 The mean mark is 27.45 and the median mark is 27.5. b) Construct the cumulative frequency table of distribution. The cumulative frequency table is shown in Step 3 of part (a). | Mark (x) | Frequency (f) | Cumulative Frequency (CF) | |:----------:|:---------------:|:-------------------------:| | 5 | 1 | 1 | | 10 | 2 | 3 | | 15 | 5 | 8 | | 20 | 12 | 20 | | 25 | 30 | 50 | | 30 | 28 | 78 | | 35 | 15 | 93 | | 40 | 5 | 98 | | 45 | 2 | 100 | c) On a graph paper, by taking 2cm to represent 5 units on the x-axis and 2cm for 10 units on the y-axis construct the cumulative frequency curve of this distribution. To construct the cumulative frequency curve (ogive): 1. Draw the x-axis (horizontal) for 'Mark' and the y-axis (vertical) for 'Cumulative Frequency'. 2. Scale the x-axis: Mark 0, 5, 10, 15, ..., 45. Each 2cm represents 5 units. 3. Scale the y-axis: Cumulative Frequency 0, 10, 20, ..., 100. Each 2cm represents 10 units. 4. Plot the points using the (Mark, Cumulative Frequency) pairs from the table: (0, 0) - (This is a common starting point for an ogive) (5, 1) (10, 3) (15, 8) (20, 20) (25, 50) (30, 78) (35, 93) (40, 98) (45, 100) 5. Draw a smooth curve connecting these plotted points. The curve should be S-shaped. Last free one today — make it count tomorrow, or type /upgrade for unlimited.