Answer :
Certainly! Let's go through a step-by-step solution based on the given frequency distribution table.
### Step-by-Step Solution
1. Identify the Midpoints of Each Bin:
To find the sample mean, we first need the midpoints of each bin. The midpoints can be calculated as the average of the lower and upper boundaries of each bin.
[tex]\[ \begin{aligned} & \text{Midpoint of } 10-20: \frac{10+20}{2} = 15 \\ & \text{Midpoint of } 20-30: \frac{20+30}{2} = 25 \\ & \text{Midpoint of } 30-40: \frac{30+40}{2} = 35 \\ & \text{Midpoint of } 40-50: \frac{40+50}{2} = 45 \\ & \text{Midpoint of } 50-60: \frac{50+60}{2} = 55 \\ & \text{Midpoint of } 60-70: \frac{60+70}{2} = 65 \\ \end{aligned} \][/tex]
So, the midpoints [tex]\( x_i \)[/tex] are: [tex]\([15, 25, 35, 45, 55, 65]\)[/tex].
2. Identify the Frequencies:
The frequencies [tex]\( f_i \)[/tex] given in the table are: [tex]\([4, 6, 5, 8, 9, 7]\)[/tex].
3. Calculate the Total Number of Observations:
The total number of observations is the sum of all frequencies.
[tex]\[ \text{Total Observations} = 4 + 6 + 5 + 8 + 9 + 7 = 39 \][/tex]
4. Calculate the Weighted Sum:
Next, we calculate the weighted sum, which involves multiplying each midpoint by its corresponding frequency and then summing the products.
[tex]\[ \begin{aligned} & 15 \times 4 = 60 \\ & 25 \times 6 = 150 \\ & 35 \times 5 = 175 \\ & 45 \times 8 = 360 \\ & 55 \times 9 = 495 \\ & 65 \times 7 = 455 \\ \end{aligned} \][/tex]
Summing these products:
[tex]\[ \text{Weighted Sum} = 60 + 150 + 175 + 360 + 495 + 455 = 1695 \][/tex]
5. Calculate the Sample Mean:
Finally, the sample mean is calculated by dividing the weighted sum by the total number of observations.
[tex]\[ \text{Sample Mean} = \frac{\text{Weighted Sum}}{\text{Total Observations}} = \frac{1695}{39} \approx 43.46 \][/tex]
### Summary of Results:
- Midpoints ([tex]\( x_i \)[/tex]): [tex]\([15, 25, 35, 45, 55, 65]\)[/tex]
- Frequencies ([tex]\( f_i \)[/tex]): [tex]\([4, 6, 5, 8, 9, 7]\)[/tex]
- Total Observations: [tex]\( 39 \)[/tex]
- Weighted Sum: [tex]\( 1695 \)[/tex]
- Sample Mean: [tex]\( \approx 43.46 \)[/tex]
These results provide us with a comprehensive understanding of the underlying distribution and central tendency of the given data.
### Step-by-Step Solution
1. Identify the Midpoints of Each Bin:
To find the sample mean, we first need the midpoints of each bin. The midpoints can be calculated as the average of the lower and upper boundaries of each bin.
[tex]\[ \begin{aligned} & \text{Midpoint of } 10-20: \frac{10+20}{2} = 15 \\ & \text{Midpoint of } 20-30: \frac{20+30}{2} = 25 \\ & \text{Midpoint of } 30-40: \frac{30+40}{2} = 35 \\ & \text{Midpoint of } 40-50: \frac{40+50}{2} = 45 \\ & \text{Midpoint of } 50-60: \frac{50+60}{2} = 55 \\ & \text{Midpoint of } 60-70: \frac{60+70}{2} = 65 \\ \end{aligned} \][/tex]
So, the midpoints [tex]\( x_i \)[/tex] are: [tex]\([15, 25, 35, 45, 55, 65]\)[/tex].
2. Identify the Frequencies:
The frequencies [tex]\( f_i \)[/tex] given in the table are: [tex]\([4, 6, 5, 8, 9, 7]\)[/tex].
3. Calculate the Total Number of Observations:
The total number of observations is the sum of all frequencies.
[tex]\[ \text{Total Observations} = 4 + 6 + 5 + 8 + 9 + 7 = 39 \][/tex]
4. Calculate the Weighted Sum:
Next, we calculate the weighted sum, which involves multiplying each midpoint by its corresponding frequency and then summing the products.
[tex]\[ \begin{aligned} & 15 \times 4 = 60 \\ & 25 \times 6 = 150 \\ & 35 \times 5 = 175 \\ & 45 \times 8 = 360 \\ & 55 \times 9 = 495 \\ & 65 \times 7 = 455 \\ \end{aligned} \][/tex]
Summing these products:
[tex]\[ \text{Weighted Sum} = 60 + 150 + 175 + 360 + 495 + 455 = 1695 \][/tex]
5. Calculate the Sample Mean:
Finally, the sample mean is calculated by dividing the weighted sum by the total number of observations.
[tex]\[ \text{Sample Mean} = \frac{\text{Weighted Sum}}{\text{Total Observations}} = \frac{1695}{39} \approx 43.46 \][/tex]
### Summary of Results:
- Midpoints ([tex]\( x_i \)[/tex]): [tex]\([15, 25, 35, 45, 55, 65]\)[/tex]
- Frequencies ([tex]\( f_i \)[/tex]): [tex]\([4, 6, 5, 8, 9, 7]\)[/tex]
- Total Observations: [tex]\( 39 \)[/tex]
- Weighted Sum: [tex]\( 1695 \)[/tex]
- Sample Mean: [tex]\( \approx 43.46 \)[/tex]
These results provide us with a comprehensive understanding of the underlying distribution and central tendency of the given data.
