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.
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Here are the calculations for both problems. Problem 1: Top Table The problem states that the mean of the distribution is 18. Step 1: Prepare the table with midpoints and f_i x_i. Let x_i be the midpoint of each class interval. | Class Interval | Frequencies (f_i) | Midpoint (x_i) | f_i x_i | |:--------------:|:-------------------:|:----------------:|:---------:| | 11-13 | 3 | 12 | 36 | | 13-15 | 6 | 14 | 84 | | 15-17 | 9 | 16 | 144 | | 17-19 | 13 | 18 | 234 | | 19-21 | f | 20 | 20f | | 21-23 | 5 | 22 | 110 | | 23-25 | 1 | 24 | 24 | | Total | 37+f | | 632+20f | Step 2: Calculate the value of frequency f. The formula for the mean of grouped data is x = ( f_i x_i)/( f_i). Given x = 18: 18 = (632 + 20f)/(37 + f) Multiply both sides by (37+f): 18(37 + f) = 632 + 20f 666 + 18f = 632 + 20f Subtract 18f from both sides and 632 from both sides: 666 - 632 = 20f - 18f 34 = 2f f = (34)/(2) The value of f is 17. Step 3: Determine the Standard Deviation. Now that f=17, the total frequency N = f_i = 37 + 17 = 54. The sum f_i x_i = 632 + 20(17) = 632 + 340 = 972. The mean x = (972)/(54) = 18. To calculate the standard deviation, we need f_i x_i^2. | Class Interval | Frequencies (f_i) | Midpoint (x_i) | x_i^2 | f_i x_i^2 | |:--------------:|:-------------------:|:----------------:|:-------:|:-----------:| | 11-13 | 3 | 12 | 144 | 432 | | 13-15 | 6 | 14 | 196 | 1176 | | 15-17 | 9 | 16 | 256 | 2304 | | 17-19 | 13 | 18 | 324 | 4212 | | 19-21 | 17 | 20 | 400 | 6800 | | 21-23 | 5 | 22 | 484 | 2420 | | 23-25 | 1 | 24 | 576 | 576 | | Total | N=54 | | | 17920 | The formula for the standard deviation (s) of grouped data is: s = sqrt(( f_i x_i^2)/(N) - x)^2 Substitute the values: s = sqrt((17920)/(54) - (18)^2) s = sqrt(331.85185... - 324) s = sqrt(7.85185...) s ≈ 2.8021 Rounding to two decimal places, the standard deviation is 2.80. --- Problem 2: QUESTION FIVE (Bottom Table) The data shows the number of times students open their WhatsApp accounts. Step 1: Prepare the table with cumulative frequencies. The total frequency N = f_i = 15+17+14+12+7 = 65. | Class Interval | Frequency (f_i) | Cumulative Frequency (CF) | |:--------------:|:-----------------:|:---------------------------:| | 1-10 | 15 | 15 | | 11-20 | 17 | 32 | | 21-30 | 14 | 46 | | 31-40 | 12 | 58 | | 41-50 | 7 | 65 | | Total | N=65 | | The formula for the k-th quartile (Q_k) of grouped data is: Q_k = L + ( (kN)/(4) - CFf ) × w where L is the lower boundary of the quartile class, CF is the cumulative frequency of the class before the quartile class, f is the frequency of the quartile class, and w is the class width. The class width for this data is 10.5 - 0.5 = 10. Step 2: Find the Lower Quartile (Q_1). The position of Q_1 is (N)/(4) = (65)/(4) = 16.25. The Q_1 class is 11-20 (since its CF=32 is the first CF 16.25). • L = 11 - 0.5 = 10.5 • CF = 15 (cumulative frequency of the class before 11-20) • f = 17 (frequency of the 11-20 class) • w = 10 Substitute these values into the formula: Q_1 = 10.5 + ( (16.25 - 15)/(17) ) × 10 Q_1 = 10.5 + ( (1.25)/(17) ) × 10 Q_1 = 10.5 + (12.5)/(17) Q_1 = 10.5 + 0.73529... Rounding to two decimal places, the lower quartile (Q_1) is 11.24. Step 3: Find the Upper Quartile (Q_3). The position of Q_3 is (3N)/(4) = (3 × 65)/(4) = (195)/(4) = 48.75. The Q_3 class is 31-40 (since its CF=58 is the first CF 48.75). • L = 31 - 0.5 = 30.5 • CF = 46 (cumulative frequency of the class before 31-40) • f = 12 (frequency of the 31-40 class) • w = 10 Substitute these values into the formula: Q_3 = 30.5 + ( (48.75 - 46)/(12) ) × 10 Q_3 = 30.5 + ( (2.75)/(12) ) × 10 Q_3 = 30.5 + (27.5)/(12) Q_3 = 30.5 + 2.2916... Rounding to two decimal places, the upper quartile (Q_3) is 32.79. Step 4: Find the Interquartile Range (IQR). The interquartile range is the difference between the upper and lower quartiles: IQR = Q_3 - Q_1 IQR = 32.79 - 11.24 IQR = 21.55 The interquartile range is 21.55. That's 2 down. 3 left today — send the next one.