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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11) The mocks obtained by 40 students in an examination are as follows: 63, 76, 87, 61, 78, 85, 77, 87, 74, 77, 80, 77, 74, 88, 72, 78, 79, 89, 85, 90, 77, 70, 81, 69, 75, 78, 73, 86, 83, 91, 69, 96, 65, 88, 84, 74, 84, 81, 83, 75. a) Copy and complete the table below using the data. Step 1: Order the data to facilitate tallying. 61, 63, 65, 69, 69, 70, 72, 73, 74, 74, 74, 75, 75, 76, 77, 77, 77, 78, 78, 78, 79, 79, 80, 81, 81, 83, 83, 84, 84, 85, 85, 86, 87, 87, 88, 88, 89, 90, 91, 96. Step 2: Calculate the frequency (f) for each class interval by tallying the data. Step 3: Calculate the class midpoint (x) for each interval using the formula: x = Lower Boundary + Upper Boundary2. Step 4: Calculate the product of frequency and midpoint (fx) for each interval. Step 5: Sum the frequencies ( f) and the fx values ( fx). The completed table is: |c|c|c|c|c| Class Boundaries & Tally & Frequency(f) & Class midpoint (x) & fx \\ 59.5 – 64.5 & II & 2 & (59.5+64.5)/(2) = 62 & 2 × 62 = 124 \\ 64.5 – 69.5 & III & 3 & (64.5+69.5)/(2) = 67 & 3 × 67 = 201 \\ 69.5 – 74.5 & IIII I & 6 & (69.5+74.5)/(2) = 72 & 6 × 72 = 432 \\ 74.5 – 79.5 & IIII IIII I & 11 & (74.5+79.5)/(2) = 77 & 11 × 77 = 847 \\ 79.5 – 84.5 & IIII II & 7 & (79.5+84.5)/(2) = 82 & 7 × 82 = 574 \\ 84.5 – 89.5 & IIII III & 8 & (84.5+89.5)/(2) = 87 & 8 × 87 = 696 \\ 89.5 – 94.5 & II & 2 & (89.5+94.5)/(2) = 92 & 2 × 92 = 184 \\ 94.5 – 99.5 & I & 1 & (94.5+99.5)/(2) = 97 & 1 × 97 = 97 \\ Total & & f=40 & & fx=3155 \\ b) Find the mean. Step 1: Use the formula for the mean of grouped data. Mean (x) = ( fx)/( f) Step 2: Substitute the values from the table. x = (3155)/(40) Step 3: Calculate the mean. x = 78.875 The mean is 78.875. c) Draw a histogram using the frequency table. To draw a histogram: • Draw a horizontal axis (x-axis) and label it "Class Boundaries" or "Mocks Obtained". Mark the class boundaries: 59.5, 64.5, 69.5, 74.5, 79.5, 84.5, 89.5, 94.5, 99.5. • Draw a vertical axis (y-axis) and label it "Frequency". Scale it from 0 to at least 11 (the maximum frequency). • For each class interval, draw a rectangular bar whose width extends from the lower class boundary to the upper class boundary, and whose height corresponds to its frequency. The bars should touch each other. 12) Two fair dice A and B, each with faces numbered 1 to 6, are thrown together. a) Construct a table showing all the equally likely outcomes. The table below shows the outcomes as (Die A, Die B) and their sum. |c|c|c|c|c|c|c| Die A \ Die B & 1 & 2 & 3 & 4 & 5 & 6 \\ 1 & (1,1)=2 & (1,2)=3