The table shows the results from the survey of a number of people. Find the value of a and b. Mean = 9.28

The total number of inductors is given as 50.

• 3 - 5: $(3+5)/2 = 4$
• 5 - 7: $(5+7)/2 = 6$
• 7 - 9: $(7+9)/2 = 8$
• 9 - 11: $(9+11)/2 = 10$
• 11 - 13: $(11+13)/2 = 12$
• 13 - 15: $(13+15)/2 = 14$

The mean ($\bar{x}$) for grouped data is given by $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$.
We are given $\bar{x} = 9.28$ and $\sum f_i = 50$.
So, $\sum f_i x_i = 9.28 \times 50 = 464$.

Now, calculate $\sum f_i x_i$:

Divide the equation by 6:

Equation 1: $a + b = 18$
Equation 2: $a + 2b = 28$

Subtract Equation 1 from Equation 2:

Substitute $b=10$ into Equation 1:

The values are: $\boxed{a=8, b=10}$

a) (ii) Determine the mode.

The modal class is the class with the highest frequency. In this case, the highest frequency is 15, which corresponds to the class interval 9 - 11.

The formula for the mode is:

Where:

• $L = 9$ (lower boundary of the modal class)
• $f_m = 15$ (frequency of the modal class)
• $f_1 = 12$ (frequency of the class preceding the modal