The marks obtained by 40 students in an examination are given. Copy and complete the frequency distribution table.

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: $x = \frac{\text{Lower Boundary} + \text{Upper Boundary}}{2}$.

Step 4: Calculate the product of frequency and midpoint (fx) for each interval.

Step 5: Sum the frequencies ($\sum f$) and the fx values ($\sum fx$).

The completed table is:

$$\begin{array}{|c|c|c|c|c|} \hline \text{Class Boundaries} & \text{Tally} & \text{Frequency(f)} & \text{Class midpoint (x)} & \text{fx} \\ \hline 59.5 – 64.5 & \text{II} & 2 & \frac{59.5+64.5}{2} = 62 & 2 \times 62 = 124 \\ \hline 64.5 – 69.5 & \text{III} & 3 & \frac{64.5+69.5}{2} = 67 & 3 \times 67 = 201 \\ \hline 69.5 – 74.5 & \text{IIII I} & 6 & \frac{69.5+74.5}{2} = 72 & 6 \times 72 = 432 \\ \hline 74.5 – 79.5 & \text{IIII IIII I} & 11 & \frac{74.5+79.5}{2} = 77 & 11 \times 77 = 847 \\ \hline 79.5 – 84.5 & \text{IIII II} & 7 & \frac{79.5+84.5}{2} = 82 & 7 \times 82 = 574 \\ \hline 84.5 – 89.5 & \text{IIII III} & 8 & \frac{84.5+89.5}{2} = 87 & 8 \times 87 = 696 \\ \hline 89.5 – 94.5 & \text{II} & 2 & \frac{89.5+94.5}{2} = 92 & 2 \times 92 = 184 \\ \hline 94.5 – 99.5 & \text{I} & 1 & \frac{94.5+99.5}{2} = 97 & 1 \times 97 = 97 \\ \hline \text{Total} & & \sum f=40 & & \sum fx=3155 \\ \hline \end{array}$$

b) Find the mean.

Step 1: Use the formula for the mean of grouped data.

$$\text{Mean} (\bar{x}) = \frac{\sum fx}{\sum f}$$

Step 2: Substitute the values from the table.

$$\bar{x} = \frac{3155}{40}$$

Step 3: Calculate the mean.

$$\bar{x} = 78.875$$

The mean is $\boxed{\text{78.875}}$.

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.

\begin{array}{|c|c|c|c|c|c|c|} \hline Die A \ Die B & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline 1 & (1,1)=2 & (1,2)=3