Question

The table below shows discrete frequency distribution data. Use it to answer the questions that follow.
Class 0-4 5-9 10-14 15-19 20-24 25-29 30-34 35-39
Frequency 5 8 10 12 7 6 3 2

Compute;
(i) Mode of the distribution (3marks)
(ii) The 7th decile (3marks)
(iii)The third quartile (4marks)

Answer 1

Answer on Question #58659 – Math – Statistics and Probability

Question

The table below shows discrete frequency distribution data. Use it to answer the questions that follow.

Compute:

(i) Mode of the distribution (3marks);

(ii) The 7th decile (3marks);

(iii) The third quartile (4marks).

Solution

(i) To find the mode of grouped distribution, the following formula will be used:

$$\mathrm{Mode} = l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \cdot h,$$

where $l$ is the lower limit of model class, $f_0$ is the frequency of class preceding, $f_1$ is the frequency of that class and $f_2$ is the frequency of class succeeding the model class respectively, $h$ is the class width.

Let's put the numbers into a table:

The mode containing class is [15-19] has the biggest frequency 12.

So the mode value is

$$\mathrm{Mode} = 15 + \frac{12 - 10}{2 \cdot 12 - 10 - 7} \cdot 4 = 15 + \frac{2}{24 - 17} \cdot 4 = 15 + \frac{8}{7} = 16 \cdot \frac{1}{7} = 16.14.$$

(ii) To find the 7th decile, we need to use the formula:

$$\mathrm {D} _ {k} = l _ {i} + \frac {\frac {k}{1 0} \cdot \sum f - f _ {D _ {k} - 1} ^ {\prime}}{f _ {D _ {k} - 1}} \cdot h,$$

where $l_{i}$ is the lower limit of decile class, $\sum f$ is the sum of the absolute frequency; $f_{D_k - 1}'$ is absolute frequency lies below the decile class; $f_{D_k - 1}$ is frequency of the decile class; $k$ is the decile number; $h$ is the class width.

The 7th decile containing class is [20-24], because Cumulative frequency in that interval is $42 > 37.1 = \frac{53}{10} \cdot 7$ .

Therefore, $\mathrm{D}_k = 20 + \frac{\frac{7}{10} \cdot 53 - 35}{7} \cdot 4 = 20 + \frac{0.7 \cdot 53 - 35}{7} \cdot 4 = 20 + \frac{2.1}{7} \cdot 4 = 20 + 1.2 = 21.2$ .

(iii) To find the third quartile, we need to use the formula:

$$Q _ {3} = l + \frac {0 . 7 5 \cdot \sum f - f _ {Q _ {3} - 1} ^ {\prime}}{f _ {Q _ {3} - 1}} \cdot h,$$

where $l$ is the lower limit of the third quartile class, $\sum f$ is the sum of the absolute frequency; $f_{Q_3 - 1}'$ is absolute frequency lies below the quartile class; $f_{Q_3 - 1}$ is frequency of the quartile class; $h$ is the class width.

The third quartile containing class is [20-24], because Cumulative frequency in that interval is $42 > 39.75 = \frac{53}{4} \cdot 3$ .

$$Q _ {3} = 2 0 + \frac {0 . 7 5 \cdot 5 3 - 3 5}{7} \cdot 4 = 2 0 + \frac {3 9 . 7 5 - 3 5}{7} \cdot 4 = 2 0 + \frac {4 . 7 5 \cdot 4}{7} = 2 0 + 2. 7 1 = 2 2. 7 1.$$

Answer: (i) 16.14; (ii) 21.2; (iii) 22.71.

www.AssignmentExpert.com