Get personalized and accurate responses to your questions with IDNLearn.com. Ask your questions and receive comprehensive, trustworthy responses from our dedicated team of experts.
Sagot :
To find the median of the given data, we follow a systematic approach for grouped data. Here's a detailed step-by-step solution:
1. Sum of all consumers:
- First, we calculate the total number of consumers:
[tex]\[ \text{Total Number of Consumers} = 4 + 9 + 22 + 42 + 56 + 64 + 68 = 265 \][/tex]
2. Cumulative Frequencies:
- Next, we determine the cumulative frequency for each class. This helps in locating the median position.
[tex]\[ \begin{align*} \text{Cumulative Frequency up to 85} &= 4 \\ \text{Cumulative Frequency up to 105} &= 4 + 9 = 13 \\ \text{Cumulative Frequency up to 125} &= 4 + 9 + 22 = 35 \\ \text{Cumulative Frequency up to 145} &= 4 + 9 + 22 + 42 = 77 \\ \text{Cumulative Frequency up to 165} &= 4 + 9 + 22 + 42 + 56 = 133 \\ \text{Cumulative Frequency up to 185} &= 4 + 9 + 22 + 42 + 56 + 64 = 197 \\ \text{Cumulative Frequency up to 205} &= 4 + 9 + 22 + 42 + 56 + 64 + 68 = 265 \\ \end{align*} \][/tex]
3. Finding the Median Position:
- The median position is found using the formula:
[tex]\[ \text{Median Position} = \frac{N}{2} = \frac{265}{2} = 132.5 \][/tex]
4. Identifying the Median Class:
- We look for the class where the cumulative frequency just exceeds the median position.
- Here, the median lies between 145 and 165, because [tex]\(132.5\)[/tex] (median position) falls in the cumulative frequency [tex]\(77\)[/tex] up until [tex]\(145\)[/tex] and [tex]\(133\)[/tex] up until [tex]\(165\)[/tex].
5. Class Interval Parameters:
[tex]\[ \text{Lower Class Boundary, } L = 145 \][/tex]
[tex]\[ \text{Frequency of Median Class, } f = 56 \][/tex]
[tex]\[ \text{Cumulative Frequency of the class before Median Class, } cf = 77 \][/tex]
[tex]\[ \text{Class Interval Width, } h = 165 - 145 = 20 \][/tex]
6. Calculating the Median:
- We use the formula for median in grouped data:
[tex]\[ \text{Median} = L + \left( \frac{\frac{N}{2} - cf}{f} \right) \times h \][/tex]
Substituting the values:
[tex]\[ \text{Median} = 145 + \left( \frac{132.5 - 77}{56} \right) \times 20 \][/tex]
[tex]\[ \text{Median} = 145 + \left( \frac{55.5}{56} \right) \times 20 \][/tex]
[tex]\[ \text{Median} = 145 + 19.82142857142857 \][/tex]
[tex]\[ \text{Median} = 164.82142857142858 \][/tex]
Therefore, the median monthly consumption is approximately [tex]\(164.82\)[/tex].
1. Sum of all consumers:
- First, we calculate the total number of consumers:
[tex]\[ \text{Total Number of Consumers} = 4 + 9 + 22 + 42 + 56 + 64 + 68 = 265 \][/tex]
2. Cumulative Frequencies:
- Next, we determine the cumulative frequency for each class. This helps in locating the median position.
[tex]\[ \begin{align*} \text{Cumulative Frequency up to 85} &= 4 \\ \text{Cumulative Frequency up to 105} &= 4 + 9 = 13 \\ \text{Cumulative Frequency up to 125} &= 4 + 9 + 22 = 35 \\ \text{Cumulative Frequency up to 145} &= 4 + 9 + 22 + 42 = 77 \\ \text{Cumulative Frequency up to 165} &= 4 + 9 + 22 + 42 + 56 = 133 \\ \text{Cumulative Frequency up to 185} &= 4 + 9 + 22 + 42 + 56 + 64 = 197 \\ \text{Cumulative Frequency up to 205} &= 4 + 9 + 22 + 42 + 56 + 64 + 68 = 265 \\ \end{align*} \][/tex]
3. Finding the Median Position:
- The median position is found using the formula:
[tex]\[ \text{Median Position} = \frac{N}{2} = \frac{265}{2} = 132.5 \][/tex]
4. Identifying the Median Class:
- We look for the class where the cumulative frequency just exceeds the median position.
- Here, the median lies between 145 and 165, because [tex]\(132.5\)[/tex] (median position) falls in the cumulative frequency [tex]\(77\)[/tex] up until [tex]\(145\)[/tex] and [tex]\(133\)[/tex] up until [tex]\(165\)[/tex].
5. Class Interval Parameters:
[tex]\[ \text{Lower Class Boundary, } L = 145 \][/tex]
[tex]\[ \text{Frequency of Median Class, } f = 56 \][/tex]
[tex]\[ \text{Cumulative Frequency of the class before Median Class, } cf = 77 \][/tex]
[tex]\[ \text{Class Interval Width, } h = 165 - 145 = 20 \][/tex]
6. Calculating the Median:
- We use the formula for median in grouped data:
[tex]\[ \text{Median} = L + \left( \frac{\frac{N}{2} - cf}{f} \right) \times h \][/tex]
Substituting the values:
[tex]\[ \text{Median} = 145 + \left( \frac{132.5 - 77}{56} \right) \times 20 \][/tex]
[tex]\[ \text{Median} = 145 + \left( \frac{55.5}{56} \right) \times 20 \][/tex]
[tex]\[ \text{Median} = 145 + 19.82142857142857 \][/tex]
[tex]\[ \text{Median} = 164.82142857142858 \][/tex]
Therefore, the median monthly consumption is approximately [tex]\(164.82\)[/tex].
Thank you for using this platform to share and learn. Don't hesitate to keep asking and answering. We value every contribution you make. Thank you for visiting IDNLearn.com. We’re here to provide accurate and reliable answers, so visit us again soon.
