To find the missing frequencies and the mean of the distribution for the given data, we follow a step-by-step process.
### Step 1: Understanding the data
We have the distribution of expenditures for 1,000 families:
- Expenditure Intervals (Rs.): 400-500, 500-600, 600-700, 700-800, 800-900
- Frequencies: 50, unknown, 450, unknown, 100
The median expenditure is given as Rs. 650, which lies in the 600-700 interval.
### Step 2: Setting up the problem
The total number of families is 1,000.
### Step 3: Calculate the total of known frequencies
The known frequencies are 50 (for 400-500), 450 (for 600-700), and 100 (for 800-900):
[tex]\[ 50 + 450 + 100 = 600 \][/tex]
### Step 4: Determine the total of the unknown frequencies
The total of the unknown frequencies (let's denote them as [tex]\(f_1\)[/tex] and [tex]\(f_2\)[/tex] for the intervals 500-600 and 700-800 respectively) needs to add up to the remaining families:
[tex]\[ 1000 - 600 = 400 \][/tex]
Therefore,
[tex]\[ f_1 + f_2 = 400 \][/tex]
### Step 5: Distribution of Median
Given that the median expenditure is Rs. 650 and it lies in the 600-700 class interval, we can infer that the missing frequencies are equally distributed. Hence, it is logical to distribute the missing families (400) equally between the two intervals:
Thus,
[tex]\[ f_1 = \frac{400}{2} = 200 \][/tex]
[tex]\[ f_2 = 400 - f_1 = 200 \][/tex]
### Step 6: Updating Frequencies
The updated frequencies are:
- 400-500: 50
- 500-600: 200
- 600-700: 450
- 700-800: 200
- 800-900: 100
### Step 7: Calculate the Mean Expenditure
To find the mean expenditure, we need the midpoint of each class interval:
- Midpoint of 400-500: 450
- Midpoint of 500-600: 550
- Midpoint of 600-700: 650
- Midpoint of 700-800: 750
- Midpoint of 800-900: 850
Now, we calculate the total expenditure by multiplying each midpoint with its corresponding frequency and summing the results:
[tex]\[
\text{Total Expenditure} = (450 \times 50) + (550 \times 200) + (650 \times 450) + (750 \times 200) + (850 \times 100)
\][/tex]
Summing these:
[tex]\[
= 22500 + 110000 + 292500 + 150000 + 85000
\][/tex]
[tex]\[
= 660000
\][/tex]
The mean expenditure is then found by dividing the total expenditure by the total number of families:
[tex]\[
\text{Mean Expenditure} = \frac{660000}{1000} = 660
\][/tex]
### Conclusion
The missing frequencies and the mean expenditure are:
- Missing frequencies: 200 for the interval 500-600 and 200 for the interval 700-800
- Mean expenditure: Rs. 660
Thus, the detailed solution to the problem is:
- [tex]\( f_1 = 200 \)[/tex]
- [tex]\( f_2 = 200 \)[/tex]
- Mean expenditure = Rs. 660
