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
To solve this problem, we first need to set up a frequency distribution table with midpoints, deviations from the assumed mean, and their products. The given data is: | Marks | No. of students (f) | |---------|---------------------| | 50-54 | 5 | | 55-59 | 8 | | 60-64 | 11 | | 65-69 | 10 | | 70-74 | 8 | | 75-79 | 4 | Note: The question states "The marks scored by 50 students", but the sum of the given frequencies is 5+8+11+10+8+4 = 46. We will proceed with f = 46 as per the provided table. We will use an assumed mean A = 62 as instructed. Step 1: Create a table to calculate f, fd, and fd^2. The midpoint (x) for each class is calculated as (lower limit + upper limit) / 2. The deviation (d) is x - A = x - 62. | Marks | f | x | d = x - 62 | fd | fd^2 | |---------|---|----|------------|------|------| | 50-54 | 5 | 52 | -10 | -50 | 500 | | 55-59 | 8 | 57 | -5 | -40 | 200 | | 60-64 | 11| 62 | 0 | 0 | 0 | | 65-69 | 10| 67 | 5 | 50 | 250 | | 70-74 | 8 | 72 | 10 | 80 | 800 | | 75-79 | 4 | 77 | 15 | 60 | 900 | | Total | f = 46 | | | fd = 100 | fd^2 = 2650 | a) (i) Mean mark Step 1: Use the formula for the mean using an assumed mean. x = A + ( fd)/( f) Step 2: Substitute the values from the table. x = 62 + (100)/(46) x = 62 + 2.173913... x = 64.173913... Step 3: Round the mean to two decimal places. x ≈ 64.17 The mean mark is 64.17. a) (ii) Standard deviation Step 1: Use the formula for standard deviation using an assumed mean. s = sqrt(( fd^2)/( f) - (( fd)/( f))^2) Step 2: Substitute the values from the table. s = sqrt((2650)/(46) - ((100)/(46))^2) s = sqrt(57.608695... - (2.173913...)^2) s = sqrt(57.608695... - 4.726086...) s = sqrt(52.882608...) s = 7.27204... Step 3: Round the standard deviation to two decimal places. s ≈ 7.27 The standard deviation is 7.27. b) Calculate the number of students who scored more than 68 marks. To find the number of students who scored more than 68 marks, we need to consider the classes above 68. The class intervals are given as discrete values (e.g., 65-69). When dealing with continuous data for interpolation, we use class boundaries. The class boundaries for 65-69 are 64.5 and 69.5.