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 is the completed table and the calculations: Completed Table: | Average pay (x) | Tally | No. of people (f) | fx | x^2 | fx^2 | | :-------------- | :---- | :---------------- | :--- | :---- | :----- | | 18 | I | 1 | 18 | 324 | 324 | | 21 | I | 1 | 21 | 441 | 441 | | 22 | III | 3 | 66 | 484 | 1452 | | 23 | II | 2 | 46 | 529 | 1058 | | 25 | II | 2 | 50 | 625 | 1250 | | 26 | I | 1 | 26 | 676 | 676 | | 27 | I | 1 | 27 | 729 | 729 | | 29 | II | 2 | 58 | 841 | 1682 | | | | f = 13 | fx = 312 | | fx^2 = 7612 | --- (a) Calculate, for this set of figures the: (i) Mean Step 1: Use the formula for the mean of grouped data. Mean () = ( fx)/( f) Step 2: Substitute the sums from the table. = (312)/(13) Step 3: Calculate the mean. = 24 The mean is 24 GHC. (ii) Median Step 1: Order the data points: 18, 21, 22, 22, 22, 23, 23, 25, 25, 26, 27, 29, 29 Step 2: Determine the position of the median. There are N=13 data points. Median position = (N+1)/(2)-th value = (13+1)/(2)-th value = 7-th value Step 3: Identify the 7th value in the ordered list. The 7th value is 23. The median is 23 GHC. (iii) Mode Step 1: Identify the value that appears most frequently in the data. From the frequency table, the value 22 appears 3 times, which is more than any other value. The mode is 22 GHC. (iv) Standard deviation Step 1: Use the formula for variance. Variance (^2) = ( fx^2)/( f) - (( fx)/( f))^2 Step 2: Substitute the sums from the table. ^2 = (7612)/(13) - ((312)/(13))^2 Step 3: Calculate the terms. ^2 = 585.5384615... - (24)^2 ^2 = 585.5384615... - 576 ^2 = 9.5384615... Step 4: Calculate the standard deviation by taking the square root of the variance. Standard Deviation () = sqrt(^2) = sqrt(9.5384615...) ≈ 3.0884 The standard deviation is 3.09 GHC (rounded to two decimal places). --- (b) If all the data sets are decreased by GHC 2, what will the following become? (i) The new mean Step 1: When a constant is subtracted from every data point, the mean also decreases by that constant. New Mean = Original Mean - 2 Step 2: Substitute the original mean. New Mean = 24 - 2 = 22 The new mean is 22 GHC. (ii) The new standard deviation Step 1: Subtracting a constant from every data point shifts the entire distribution but does not change its spread or variability. Therefore, the standard deviation remains the same. New Standard Deviation = Original Standard Deviation Step 2: Substitute the original standard deviation. New Standard Deviation ≈ 3.09 The new standard deviation is 3.09 GHC.