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's the solution to the problem: First, we extend the given frequency table to include midpoints (x), fx, and cumulative frequency (C_f). | Marks | Class Boundaries | Frequency (f) | Midpoint (x) | fx | Cumulative Frequency (C_f) | | :------ | :--------------- | :-------------- | :------------- | :---- | :--------------------------- | | 40-44 | 39.5-44.5 | 6 | 42 | 252 | 6 | | 45-49 | 44.5-49.5 | 4 | 47 | 188 | 10 | | 50-54 | 49.5-54.5 | 7 | 52 | 364 | 17 | | 55-59 | 54.5-59.5 | 6 | 57 | 342 | 23 | | 60-64 | 59.5-64.5 | 12 | 62 | 744 | 35 | | 65-69 | 64.5-69.5 | 4 | 67 | 268 | 39 | | 70-74 | 69.5-74.5 | 5 | 72 | 360 | 44 | | 75-79 | 74.5-79.5 | 3 | 77 | 231 | 47 | | 80-84 | 79.5-84.5 | 3 | 82 | 246 | 50 | | Total | | f = 50 | | fx = 2995 | | a) State the modal class Step 1: Identify the highest frequency. The highest frequency in the table is 12. Step 2: Identify the class corresponding to the highest frequency. The class interval corresponding to the frequency 12 is 60-64. The modal class is 60-64. b) Estimate: i) The mean mark Step 1: Use the formula for the mean of grouped data. The mean (x) is given by the formula: x = ( fx)/( f) Step 2: Substitute the values from the table. From the table, fx = 2995 and f = 50. x = (2995)/(50) Step 3: Calculate the mean. x = 59.9 The estimated mean mark is 59.9. ii) The median Step 1: Determine the position of the median. The total number of students (N) is 50. The position of the median is (N)/(2) = (50)/(2) = 25^th value. Step 2: Identify the median class. From the cumulative frequency column, the 25^th value falls within the class 60-64 (since the cumulative frequency for 55-59 is 23, and for 60-64 it is 35). So, the median class is 60-64. Step 3: Identify the values for the median formula. • Lower class boundary of the median class (L) = 59.5 • Total frequency (N) = 50 • Cumulative frequency of the class before the median class (C_f) = 23 (for class 55-59) • Frequency of the median class (f_m) = 12 • Class width (c) = 64.5 - 59.5 = 5 Step 4: Use the formula for the median of grouped data. Median = L + ((N)/(2) - C_ff_m)c Step 5: Substitute the values and calculate the median. Median = 59.5 + ((50)/(2) - 2312)5 Median = 59.5 + ((25 - 23)/(12))5 Median = 59.5 + ((2)/(12))5 Median = 59.5 + ((1)/(6))5 Median = 59.5 + (5)/(6) Median = 59.5 + 0.8333... Median ≈ 60.33 The estimated median is 60.33. Last free one today — make it count tomorrow, or type /upgrade for unlimited.