Question 1

We will use the direct method to find the mean number of plants per house.

Step 1 — Calculate Class Marks and Products

First, we find the class mark ($x_i$) for each interval. The class mark is the midpoint of the interval. Then, we calculate the product of frequency ($f_i$) and class mark ($x_i$).

| Number of plants | Number of houses ($f_i$) | Class mark ($x_i$) | $f_i x_i$ | | :--------------- | :----------------------- | :----------------- | :-------- | | 0 - 2 | 1 | 1 | 1 | | 2 - 4 | 2 | 3 | 6 | | 4 - 6 | 1 | 5 | 5 | | 6 - 8 | 5 | 7 | 35 | | 8 - 10 | 6 | 9 | 54 | | 10 - 12 | 2 | 11 | 22 | | 12 - 14 | 3 | 13 | 39 | | Total | 20 | | 162 |

Let's find the sum of frequencies.

$\sum f_i = 1 + 2 + 1 + 5 + 6 + 2 + 3$

$\boxed{\sum f_i = 20}$

Now, let's find the sum of $f_i x_i$.

$\sum f_i x_i = 1 + 6 + 5 + 35 + 54 + 22 + 39$

$\boxed{\sum f_i x_i = 162}$

Step 2 — Calculate the Mean

We use the formula for the mean of grouped data. The mean ($\bar{x}$) is the sum of $f_i x_i$ divided by the sum of $f_i$.

$\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$

$\bar{x} = \frac{162}{20}$

$\bar{x} = 8.1$

$\boxed{\text{Mean number of plants} = 8.1}$

Answer

(i) The mean number of plants per house is 8.1. (ii) We used the Direct Method for finding the mean. (iii) This method was chosen because the values of the class marks ($x_i$) and frequencies ($f_i$) were small. This makes calculations straightforward and less prone to errors.