Question: Section 3: Analysis of Data |Sub Strand 4.5|
Task 3.1: Parameters from Grouped Data
Using the frequency table, you completed in Task 2.2, estimate the statistical parameters for the grouped data.
1. Identify the Modal Class: (1 mark)
2. Calculate the Estimated Mean: Use the class midpoints ($x$) to find $\bar{x} = \frac{\sum f x}{\sum f}$. Show your table extensions and final steps below.
(1 mark)
1. **State the problem:**
We are given grouped data with frequencies and need to find:
- The modal class (the class interval with the highest frequency).
- The estimated mean using class midpoints.
2. **Identify the Modal Class:**
The modal class is the class interval with the highest frequency $f$.
3. **Calculate the Estimated Mean:**
- First, find the midpoint $x$ of each class interval: $$x = \frac{\text{lower limit} + \text{upper limit}}{2}$$
- Then multiply each midpoint by its frequency $f$ to get $f x$.
- Sum all frequencies: $$\sum f$$
- Sum all $f x$: $$\sum f x$$
- Use the formula for the estimated mean:
$$\bar{x} = \frac{\sum f x}{\sum f}$$
4. **Example Table Extension:**
| Class Interval | Frequency ($f$) | Midpoint ($x$) | $f x$ |
|----------------|-----------------|----------------|-------|
| $a_1$-$b_1$ | $f_1$ | $x_1 = \frac{a_1 + b_1}{2}$ | $f_1 x_1$ |
| $a_2$-$b_2$ | $f_2$ | $x_2 = \frac{a_2 + b_2}{2}$ | $f_2 x_2$ |
| ... | ... | ... | ... |
| $a_n$-$b_n$ | $f_n$ | $x_n = \frac{a_n + b_n}{2}$ | $f_n x_n$ |
5. **Calculate sums:**
$$\sum f = f_1 + f_2 + \cdots + f_n$$
$$\sum f x = f_1 x_1 + f_2 x_2 + \cdots + f_n x_n$$
6. **Calculate the estimated mean:**
$$\bar{x} = \frac{\sum f x}{\sum f}$$
7. **Explain simplification with cancellation:**
If possible, simplify the fraction by canceling common factors:
$$\bar{x} = \frac{\cancel{c} m}{\cancel{c} n} = \frac{m}{n}$$
8. **Final answer:**
The modal class is the class with the highest frequency.
The estimated mean is the value of $\bar{x}$ calculated above.
This method provides an estimate of the mean for grouped data using class midpoints and frequencies.