Question

The weekly observations on the cost of living index in a certain city for the year 2004-2005 are given below. Compute the weekly cost of living index.

Cost of living index	Number of students
$1400 - 1500$	$5$
$1500 - 1600$	$10$
$1600 - 1700$	$20$
$1700 - 1800$	$9$
$1800 - 1900$	$6$
$1900 - 2000$	$2$

Answer 1

According to given in the question we have to determine the weakly cost of living index which can be determine with the help of the mean so, first of all we have to understand about the mean which is as explained below:
Mean: The mean or we can say that the average of a data set which can be found by adding all the given numbers in the data set and then dividing by the number of the values in the set.
Now, we have to obtain the mean of the cost of living index which is basically ${x_i}$ for the given table with the help of the mean which is as explained above.
Now, we have to determine the product of ${x_i}$ and the given number of students which is the frequency for the given data is ${f_i}$
Now, we have to determine the sum of all obtained $\sum {{f_i}{x_i}}$ and $\sum {{f_i}}$ after that we have to determine the mean with the help of the formula as mentioned below:

Formula used: Mean$= \dfrac{{\sum {{f_i}{x_i}} }}{{\sum {{f_i}} }}............(A)$
So, with the help of the formula (A) above, we can determine the weekly cost of living index.

Complete step-by-step solution:
Step 1: First of all we have to obtain the mean of the cost of living index which is basically ${x_i}$for the given table with the help of the mean which is as explained in the solution hint. Hence,

Cost of living index	Number of students$({f_i})$	$({x_i})$
$1400 - 1500$	$5$	$\dfrac{{1400 + 1500}}{2} = 1450$
$1500 - 1600$	$10$	$\dfrac{{1500 + 1600}}{2} = 1550$
$1600 - 1700$	$20$	$\dfrac{{1600 + 1700}}{2} = 1650$
$1700 - 1800$	$9$	$\dfrac{{1700 + 1800}}{2} = 1750$
$1800 - 1900$	$6$	$\dfrac{{1800 + 1900}}{2} = 1850$
$1900 - 2000$	$2$	$\dfrac{{1900 + 2000}}{2} = 1950$

Step 2: Now, we have to determine the product of $({f_i})$and $({x_i})$as from the table obtained in the solution step 1. Hence,

Cost of living index	Number of students$({f_i})$	$({x_i})$	$({f_i}{x_i})$
$1400 - 1500$	$5$	$\dfrac{{1400 + 1500}}{2} = 1450$	$5 \times 1450 = 7250$
$1500 - 1600$	$10$	$\dfrac{{1500 + 1600}}{2} = 1550$	$10 \times 1550 = 15500$
$1600 - 1700$	$20$	$\dfrac{{1600 + 1700}}{2} = 1650$	$20 \times 1650 = 33000$
$1700 - 1800$	$9$	$\dfrac{{1700 + 1800}}{2} = 1750$	$9 \times 1750 = 15750$
$1800 - 1900$	$6$	$\dfrac{{1800 + 1900}}{2} = 1850$	$6 \times 1850 = 11100$
$1900 - 2000$	$2$	$\dfrac{{1900 + 2000}}{2} = 1950$	$2 \times 1950 = 3900$

Step 3: Now, we have to determine the sum of all $({f_i}{x_i})$as obtained in the table in the solution step 2. Hence,

$$\Rightarrow \sum {({f_i}{x_i}) = 7250 + 15500 + 33000 + 15750 + 11100 + 3900} \\\ \Rightarrow \sum {({f_i}{x_i})} = 86500.............(1) \\\$$

Step 4: Now, we have to determine the sum of all $({f_i})$as obtained in the table in the solution step 2. Hence,

$$\Rightarrow \sum {({f_i}) = 5 + 10 + 20 + 9 + 6 + 2} \\\ \Rightarrow \sum {({f_i}) = 52................(2)} \\\$$

Step 5: Now, we have to determine the mean of the data with the help of the formula (A) as mentioned in the solution hint. Hence,
Mean:
$\Rightarrow \dfrac{{86500}}{{52}} = 1663.46$

Hence, with the help of the formula (A) as mentioned in the solution hint we have determined the mean which is the weakly cost of living index is 1663.46.

Note: Mean is basically the same as the average for the given data set in which first of all we have to determine the sum of all the given numbers or data and then we have to divide the sum by the total number of data sets.
To obtain the weakly cost of living index basically we have to determine the mean.