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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You're on a roll — here are the solutions to the questions: 2) University Admissions Pie Chart: To draw a pie chart, we first calculate the total number of students and the angle for each faculty. Total students = 350 (Education) + 150 (Medicine) + 200 (Engineering) + 100 (Law) + 100 (Arts) = 900 students. The angle for each faculty is calculated as: Angle = Number of students in facultyTotal number of students × 360^ • Education: (350)/(900) × 360^ = 140^ • Medicine: (150)/(900) × 360^ = 60^ • Engineering: (200)/(900) × 360^ = 80^ • Law: (100)/(900) × 360^ = 40^ • Arts: (100)/(900) × 360^ = 40^ To draw the pie chart, use a compass to draw a circle with a radius of 5 cm. Then, use a protractor to mark out the calculated angles from the center of the circle for each faculty. Label each sector accordingly. Variance of Ages Distribution: To calculate the variance for grouped data, we first find the midpoint (x) for each class interval, then calculate fx and fx^2. Step 1: Create a table with midpoints, fx, and fx^2. | Ages (yrs) | Frequency (f) | Midpoint (x) | fx | x^2 | fx^2 | | :--------- | :-------------- | :------------- | :--- | :---- | :----- | | 21 – 30 | 3 | 25.5 | 76.5 | 650.25 | 1950.75 | | 31 – 40 | 10 | 35.5 | 355 | 1260.25 | 12602.5 | | 41 – 50 | 14 | 45.5 | 637 | 2070.25 | 28983.5 | | 51 – 60 | 10 | 55.5 | 555 | 3080.25 | 30802.5 | | 61 – 70 | 8 | 65.5 | 524 | 4290.25 | 34322 | | Total | f = 45 | | fx = 2147.5 | | fx^2 = 108661.25 | Step 2: Calculate the mean (x). x = ( fx)/( f) = (2147.5)/(45) ≈ 47.7222 Step 3: Calculate the variance (^2). ^2 = ( fx^2)/( f) - x^2 ^2 = (108661.25)/(45) - (47.7222)^2 ^2 = 2414.6944 - 2277.4081 ^2 = 137.2863 Rounding to two decimal places: Variance = 137.29 3) The table shows the height of teak trees harvested by a farmer: | Height (m) (x) | Number of trees (f) | | :--------------- | :-------------------- | | 3 | 4 | | 4 | 6 | | 5 | 4 | | 6 | 5 | | 7 | 6 | | 8 | 2 | a) Find the median height. Step 1: Find the total number of trees ( f). f = 4 + 6 + 4 + 5 + 6 + 2 = 27 Step 2: Determine the position of the median. Median position = ( f + 1)/(2) = (27 + 1)/(2) = 28