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 calculate the mean and standard deviation from the given frequency distribution, we will use the following formulas: Mean (x) x = ( fx)/( f) Standard Deviation () = sqrt(( fx^2)/( f) - (( fx)/( f))^2) First, let's create a table to organize the calculations for fx and fx^2. The total number of students is f = 50. | Hours per day (x) | Number of students (f) | fx | x^2 | fx^2 | |---------------------|--------------------------|------|-------|--------| | 4 | 5 | 20 | 16 | 80 | | 5 | 7 | 35 | 25 | 175 | | 6 | 5 | 30 | 36 | 180 | | 7 | 9 | 63 | 49 | 441 | | 8 | 12 | 96 | 64 | 768 | | 9 | 4 | 36 | 81 | 324 | | 10 | 3 | 30 | 100 | 300 | | 11 | 5 | 55 | 121 | 605 | | Total | 50 | 365 | | 2873 | From the table, we have: f = 50 fx = 365 fx^2 = 2873 a) Mean Step 1: Use the formula for the mean. x = ( fx)/( f) Step 2: Substitute the calculated values into the formula. x = (365)/(50) Step 3: Calculate the mean. x = 7.3 The mean is 7.30 hours (correct to two decimal places). Mean = 7.30 b) Standard Deviation Step 1: Use the formula for the standard deviation. = sqrt(( fx^2)/( f) - (( fx)/( f))^2) Step 2: Substitute the calculated values into the formula. = sqrt((2873)/(50) - ((365)/(50))^2) Step 3: Perform the division and squaring. = sqrt(57.46 - (7.3)^2) = sqrt(57.46 - 53.29) Step 4: Subtract the values. = sqrt(4.17) Step 5: Calculate the square root and round to two decimal places. ≈ 2.04205778 ≈ 2.04 The standard deviation is 2.04 hours (correct to two decimal places). Standard Deviation = 2.04 That's 2 down. 3 left today — send the next one.