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 sample standard deviation (S.D.) for grouped data, we use the formula: s = sqrt( f_i (x_i - x))^2N-1 where: s is the sample standard deviation f_i is the frequency of each class x_i is the midpoint of each class x is the sample mean N is the total number of observations ( f_i) First, we need to determine the midpoints (x_i) for each class. Given the classes "700-799" and "800-899", we assume a consistent class width of 100 for the other single-value classes (e.g., "900" means "900-999"). Here is the table with midpoints and calculations: | Class | Frequency (f_i) | Midpoint (x_i) | f_i x_i | (x_i - x) | (x_i - x)^2 | f_i (x_i - x)^2 | | :---------- | :---------------- | :--------------- | :-------- | :----------------- | :------------------- | :----------------------- | | 700-799 | 4 | 749.5 | 2998 | -500 | 250000 | 1000000 | | 800-899 | 7 | 849.5 | 5946.5 | -400 | 160000 | 1120000 | | 900-999 | 8 | 949.5 | 7596 | -300 | 90000 | 720000 | | 1000-1099 | 10 | 1049.5 | 10495 | -200 | 40000 | 400000 | | 1100-1199 | 12 | 1149.5 | 13794 | -100 | 10000 | 120000 | | 1200-1299 | 17 | 1249.5 | 21241.5 | 0 | 0 | 0 | | 1300-1399 | 13 | 1349.5 | 17543.5 | 100 | 10000 | 130000 | | 1400-1499 | 10 | 1449.5 | 14495 | 200 | 40000 | 400000 | | 1500-1599 | 9 | 1549.5 | 13945.5 | 300 | 90000 | 810000 | | 1600-1699 | 7 | 1649.5 | 11546.5 | 400 | 160000 | 1120000 | | 1700-1799 | 2 | 1749.5 | 3499 | 500 | 250000 | 500000 | | 1800-1899 | 1 | 1849.5 | 1849.5 | 600 | 360000 | 360000 | | Total | 100 | | 124950| | | 6680000 | Step 1: Calculate the total number of observations (N). N = f_i = 4+7+8+10+12+17+13+10+9+7+2+1 = 100 Step 2: Calculate the sum of f_i x_i. f_i x_i = 2998 + 5946.5 + 7596 + 10495 + 13794 + 21241.5 + 17543.5 + 14495 + 13945.5 + 11546.5 + 3499 + 1849.5 = 124950 Step 3: Calculate the sample mean (x). x = ( f_i x_i)/(N) = (124950)/(100) = 1249.5 Step 4: Calculate the sum of f_i (x_i - x)^2. f_i (x_i - x)^2 = 1000000 + 1120000 + 720000 + 400000 + 120000 + 0 + 130000 + 400000 + 810000 + 1120000 + 500000 + 360000 = 6680000 Step 5: Calculate the sample standard deviation (s). s = sqrt( f_i (x_i - x))^2N-1 s = sqrt((6680000)/(100-1)) s = sqrt((6680000)/(99)) s = sqrt(67474.74747...) s ≈ 259.75905 Rounding to two decimal places, the sample standard deviation is 259.76. s ≈ 259.76 3 done, 2 left today. You're making progress.