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 calculations for the given data. First, let's organize the data and calculate the necessary values for the mean and standard deviation. Step 1: Create a table to find class marks (x), fx, and fx^2. The class mark (x) for each interval is the midpoint of the class. The total frequency f = 18 + 34 + 37 + 11 = 100. | Class Interval | Frequency (f) | Class Mark (x) | fx | x^2 | fx^2 | | :------------- | :-------------- | :--------------- | :--- | :---- | :----- | | 25-28 | 18 | (25+28)/(2) = 26.5 | 18 × 26.5 = 477 | 26.5^2 = 702.25 | 18 × 702.25 = 12640.5 | | 29-32 | 34 | (29+32)/(2) = 30.5 | 34 × 30.5 = 1037 | 30.5^2 = 930.25 | 34 × 930.25 = 31628.5 | | 33-36 | 37 | (33+36)/(2) = 34.5 | 37 × 34.5 = 1276.5 | 34.5^2 = 1190.25 | 37 × 1190.25 = 44039.25 | | 37-40 | 11 | (37+40)/(2) = 38.5 | 11 × 38.5 = 423.5 | 38.5^2 = 1482.25 | 11 × 1482.25 = 16304.75 | | Total | f = 100 | | fx = 3214 | | fx^2 = 104613 | Step 2: Calculate the mean using class mark (for part ii). The formula for the mean (x) of grouped data using class marks is: x = ( fx)/( f) Substitute the values from the table: x = (3214)/(100) x = 32.14 ii) The mean using class mark is: 32.14 Step 3: Calculate the standard deviation (for part i). The formula for the standard deviation () of grouped data is: = sqrt(( fx^2)/( f) - (( fx)/( f))^2) We already calculated ( fx)/( f) = x = 32.14. Substitute the values from the table: = sqrt((104613)/(100) - (32.14)^2) = sqrt(1046.13 - 1032.98 ) = sqrt(13.15) ≈ 3.626 i) The standard deviation is: 3.63 (rounded to two decimal places) Step 4: Create a table to find deviations (d) and fd using the assumed mean (for part iii). The question states "given that mean is 30.5". We will use A = 30.5 as the assumed mean. The deviation d = x - A. | Class Interval | Frequency (f) | Class Mark (x) | d = x - 30.5 | fd | | :------------- | :-------------- | :--------------- | :------------- | :--- | | 25-28 | 18 | 26.5 | 26.5 - 30.5 = -4 | 18 × (-4) = -72 | | 29-32 | 34 | 30.5 | 30.5 - 30.5 = 0 | 34 × 0 = 0 | | 33-36 | 37 | 34.5 | 34.5 - 30.5 = 4 | 37 × 4 = 148 | | 37-40 | 11 | 38.5 | 38.5 - 30.5 = 8 | 11 × 8 = 88 | | Total | f = 100 | | | fd = 164 | Step 5: Calculate the mean using the assumed mean method (for part iii). The formula for the mean (x) using the assumed mean method is: x = A + ( fd)/( f) Substitute the values: x = 30.5 + (164)/(100) x = 30.5 + 1.64 x = 32.14 iii) The mean using the assumed mean method is: 32.14 That's 2 down. 3 left today — send the next one.