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 are the solutions to the problems. Problem a): Find the mean, median, and mode. First, let's list the data from the table: Marks Scored (x): 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 Frequency (f): 2, 3, 4, 4, 4, 4, 5, 3, 2, 1 The total number of students is f = 2+3+4+4+4+4+5+3+2+1 = 32. The sum of (fx) is fx = (1 × 2) + (2 × 3) + (3 × 4) + (4 × 4) + (5 × 4) + (6 × 4) + (7 × 5) + (8 × 3) + (9 × 2) + (10 × 1) fx = 2 + 6 + 12 + 16 + 20 + 24 + 35 + 24 + 18 + 10 = 167. Mean: Step 1: Use the formula for the mean of a frequency distribution. Mean = ( fx)/( f) Step 2: Substitute the calculated values. Mean = (167)/(32) Mean = 5.21875 The mean is 5.21875. Median: Step 1: Determine the position of the median. The total number of data points is N = f = 32. Since N is even, the median is the average of the (N)/(2)-th and ((N)/(2)+1)-th values. The positions are (32)/(2) = 16th and (32)/(2)+1 = 17th. Step 2: Find the cumulative frequencies to locate the values. | Marks Scored (x) | Frequency (f) | Cumulative Frequency (CF) | | :----------------- | :-------------- | :------------------------ | | 1 | 2 | 2 | | 2 | 3 | 5 | | 3 | 4 | 9 | | 4 | 4 | 13 | | 5 | 4 | 17 | | 6 | 4 | 21 | | 7 | 5 | 26 | | 8 | 3 | 29 | | 9 | 2 | 31 | | 10 | 1 | 32 | The 16th value falls in the '4 marks' category (CF up to 13, next is 17). So the 16th value is 4. The 17th value falls in the '5 marks' category (CF up to 17). So the 17th value is 5. Step 3: Calculate the median. Median = 16th value + 17th value2 Median = (4+5)/(2) Median = (9)/(2) Median = 4.5 The median is 4.5. Mode: Step 1: Identify the mark with the highest frequency. Looking at the frequency column, the highest frequency is 5, which corresponds to a mark of 7. The mode is 7. Problem b): Percentage of students who scored at least 8 marks. Step 1: Find the number of students who scored at least 8 marks. "At least 8 marks" means students who scored 8, 9, or 10 marks. Number of students with 8 marks = 3 Number of students with 9 marks = 2 Number of students with 10 marks = 1 Total students with at least 8 marks = 3 + 2 + 1 = 6. Step 2: Calculate the percentage. Percentage = Number of students with at least 8 marksTotal number of students × 100\% Percentage = (6)/(32) × 100\% Percentage = 0.1875 × 100\% Percentage = 18.75\% The percentage of students who scored at least 8 marks is 18.75\%. That's 2 down. 3 left today — send the next one.