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 solution for Question 2: The table shows the distribution of marks obtained by 40 students: | Marks (x) | Number of students (f) | | :-------- | :--------------------- | | 1-5 | 2 | | 6-10 | 4 | | 11-15 | 10 | | 16-20 | 12 | | 21-25 | 8 | | 26-30 | 4 | a) Calculate the mean mark. Step 1: Determine the midpoint (x) for each class interval and calculate fx. | Marks | f | Midpoint (x) | fx | | :---- | :- | :----------- | :-- | | 1-5 | 2 | (1+5)/(2) = 3 | 2 × 3 = 6 | | 6-10 | 4 | (6+10)/(2) = 8 | 4 × 8 = 32 | | 11-15 | 10 | (11+15)/(2) = 13 | 10 × 13 = 130 | | 16-20 | 12 | (16+20)/(2) = 18 | 12 × 18 = 216 | | 21-25 | 8 | (21+25)/(2) = 23 | 8 × 23 = 184 | | 26-30 | 4 | (26+30)/(2) = 28 | 4 × 28 = 112 | Step 2: Calculate the sum of frequencies ( f) and the sum of fx ( fx). f = 2+4+10+12+8+4 = 40 fx = 6+32+130+216+184+112 = 680 Step 3: Apply the formula for the mean of grouped data. Mean (x) = ( fx)/( f) x = (680)/(40) x = 17 b) Construct a cumulative frequency table and use it to determine the median mark. Step 1: Construct the cumulative frequency table, including class boundaries. | Marks | f | Class Boundaries | Cumulative Frequency (cf) | | :---- | :- | :--------------- | :------------------------ | | 1-5 | 2 | 0.5 - 5.5 | 2 | | 6-10 | 4 | 5.5 - 10.5 | 2+4=6 | | 11-15 | 10 | 10.5 - 15.5 | 6+10=16 | | 16-20 | 12 | 15.5 - 20.5 | 16+12=28 | | 21-25 | 8 | 20.5 - 25.5 | 28+8=36 | | 26-30 | 4 | 25.5 - 30.5 | 36+4=40 | Step 2: Determine the median position and identify the median class. Total frequency N = 40. The median position is (N)/(2) = (40)/(2) = 20^th value. The cumulative frequency first exceeds 20 in the class 16-20 (cf = 28). So, the median class is 16-20. Step 3: Identify the values for the median formula. The median formula for grouped data is: Median = L + ((N)/(2) - cf_bf_m) × w Where: L = Lower class boundary of the median class = 15.5 N = Total frequency = 40 cf_b = Cumulative frequency of the class before the median class = 16 f_m = Frequency of the median class = 12 w = Class width = 5.5 - 0.5 = 5 Step 4: Apply the median formula. Median = 15.5 + ((40)/(2) - 1612) × 5 Median = 15.5 + ((20 - 16)/(12)) × 5 Median = 15.5 + ((4)/(12)) × 5 Median = 15.5 + (1)/(3) × 5 Median = 15.5 + (5)/(3) Median = 15.5 + 1.666... Median ≈ 17.17 c) Determine the mode mark. Step 1: Identify the modal class. The modal class is the class with the highest frequency. From the table, the highest frequency is 12, which corresponds to the class interval 16-20. So, the modal class is 16-20. Step 2: Identify the values for the mode formula. The mode formula for grouped data is: Mode = L + ((f_m - f_1)/((f_m - f_1) + (f_m - f_2))) × w Where: L = Lower class boundary of the modal class = 15.5 f_m = Frequency of the modal class = 12 f_1 = Frequency of the class before the modal class (11-15) = 10 f_2 = Frequency of the class after the modal class (21-25) = 8 w = Class width = 5 Step 3: Apply the mode formula. Mode = 15.5 + ((12 - 10)/((12 - 10) + (12 - 8))) × 5 Mode = 15.5 + ((2)/(2 + 4)) × 5 Mode = 15.5 + ((2)/(6)) × 5 Mode = 15.5 + (1)/(3) × 5 Mode = 15.5 + (5)/(3) Mode = 15.5 + 1.666... Mode ≈ 17.17 What's next?