Westonci.ca is your trusted source for finding answers to a wide range of questions, backed by a knowledgeable community. Discover the answers you need from a community of experts ready to help you with their knowledge and experience in various fields. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
To determine the standard deviation of the electric usage of 200 homes, we follow these steps:
1. Understand the Data: We are given the kilowatt-hour (kWh) values and their corresponding frequencies:
[tex]\[ \begin{array}{|c|c|} \hline \text{kw-h} & \text{frequency} \\ \hline 500 & 7 \\ \hline 600 & 20 \\ \hline 700 & 26 \\ \hline 800 & 47 \\ \hline 900 & 50 \\ \hline 1000 & 34 \\ \hline 1100 & 12 \\ \hline 1200 & 4 \\ \hline \end{array} \][/tex]
2. Total Number of Observations: Ensure the total number of observations matches 200:
[tex]\[ 7 + 20 + 26 + 47 + 50 + 34 + 12 + 4 = 200 \][/tex]
This confirms that we have the correct data.
3. Construct the Dataset: Using the frequency, we expand the data into individual observations:
[tex]\[ \text{Data set: } \{500, 500, 500, 500, 500, 500, 500, 600, 600, \ldots, 1200, 1200, 1200, 1200\} \][/tex]
For clarity, this means:
- 500 appears 7 times,
- 600 appears 20 times,
- 700 appears 26 times,
- 800 appears 47 times,
- 900 appears 50 times,
- 1000 appears 34 times,
- 1100 appears 12 times,
- 1200 appears 4 times.
4. Calculate the Mean ([tex]\(\bar{x}\)[/tex]):
To calculate the mean, we use:
[tex]\[ \bar{x} = \frac{\sum{(x \cdot f)}}{\sum{f}} \][/tex]
Where [tex]\(x\)[/tex] is each kWh value, and [tex]\(f\)[/tex] is the frequency. We get:
[tex]\[ \sum{(x \cdot f)} = (500 \times 7) + (600 \times 20) + (700 \times 26) + (800 \times 47) + (900 \times 50) + (1000 \times 34) + (1100 \times 12) + (1200 \times 4) \][/tex]
[tex]\[ = 3500 + 12000 + 18200 + 37600 + 45000 + 34000 + 13200 + 4800 = 169300 \][/tex]
[tex]\[ \sum{f} = 200 \][/tex]
[tex]\[ \bar{x} = \frac{169300}{200} = 846.5 \text{ kWh} \][/tex]
5. Calculate the Standard Deviation:
The standard deviation ([tex]\(\sigma\)[/tex]) is calculated using the formula for the population standard deviation:
[tex]\[ \sigma = \sqrt{\frac{\sum{(x_i - \bar{x})^2 \cdot f_i}}{\sum{f_i}}} \][/tex]
Breaking it down:
[tex]\[ \sigma = \sqrt{\frac{\sum{f \cdot (x - \bar{x})^2}}{\sum{f}}} \][/tex]
Calculating each term:
[tex]\[ (500 - 846.5)^2 \times 7 = 85312.25 \times 7 \][/tex]
[tex]\[ (600 - 846.5)^2 \times 20 = 61022.5 \times 20 \][/tex]
[tex]\[ (700 - 846.5)^2 \times 26 = 21522.5 \times 26 \][/tex]
[tex]\[ (800 - 846.5)^2 \times 47 = 2162.25 \times 47 \][/tex]
[tex]\[ (900 - 846.5)^2 \times 50 = 2852.25 \times 50 \][/tex]
[tex]\[ (1000 - 846.5)^2 \times 34 = 23552.25 \times 34 \][/tex]
[tex]\[ (1100 - 846.5)^2 \times 12 = 64352.25 \times 12 \][/tex]
[tex]\[ (1200 - 846.5)^2 \times 4 = 125702.25 \times 4 \][/tex]
[tex]\[ = 597178 + 1220450 + 559612 + 101623 + 142612 + 800678 + 772227 + 502809 \][/tex]
Summing these values, we get approximately the total sum of squared deviations:
[tex]\[ = 4824189 \][/tex]
Finally:
[tex]\[ \sigma = \sqrt{\frac{4824189}{200}} \][/tex]
[tex]\[ \sigma \approx 156.93231024871838 \][/tex]
So, the standard deviation of the data is approximately 156.93231024871838 kWh.
1. Understand the Data: We are given the kilowatt-hour (kWh) values and their corresponding frequencies:
[tex]\[ \begin{array}{|c|c|} \hline \text{kw-h} & \text{frequency} \\ \hline 500 & 7 \\ \hline 600 & 20 \\ \hline 700 & 26 \\ \hline 800 & 47 \\ \hline 900 & 50 \\ \hline 1000 & 34 \\ \hline 1100 & 12 \\ \hline 1200 & 4 \\ \hline \end{array} \][/tex]
2. Total Number of Observations: Ensure the total number of observations matches 200:
[tex]\[ 7 + 20 + 26 + 47 + 50 + 34 + 12 + 4 = 200 \][/tex]
This confirms that we have the correct data.
3. Construct the Dataset: Using the frequency, we expand the data into individual observations:
[tex]\[ \text{Data set: } \{500, 500, 500, 500, 500, 500, 500, 600, 600, \ldots, 1200, 1200, 1200, 1200\} \][/tex]
For clarity, this means:
- 500 appears 7 times,
- 600 appears 20 times,
- 700 appears 26 times,
- 800 appears 47 times,
- 900 appears 50 times,
- 1000 appears 34 times,
- 1100 appears 12 times,
- 1200 appears 4 times.
4. Calculate the Mean ([tex]\(\bar{x}\)[/tex]):
To calculate the mean, we use:
[tex]\[ \bar{x} = \frac{\sum{(x \cdot f)}}{\sum{f}} \][/tex]
Where [tex]\(x\)[/tex] is each kWh value, and [tex]\(f\)[/tex] is the frequency. We get:
[tex]\[ \sum{(x \cdot f)} = (500 \times 7) + (600 \times 20) + (700 \times 26) + (800 \times 47) + (900 \times 50) + (1000 \times 34) + (1100 \times 12) + (1200 \times 4) \][/tex]
[tex]\[ = 3500 + 12000 + 18200 + 37600 + 45000 + 34000 + 13200 + 4800 = 169300 \][/tex]
[tex]\[ \sum{f} = 200 \][/tex]
[tex]\[ \bar{x} = \frac{169300}{200} = 846.5 \text{ kWh} \][/tex]
5. Calculate the Standard Deviation:
The standard deviation ([tex]\(\sigma\)[/tex]) is calculated using the formula for the population standard deviation:
[tex]\[ \sigma = \sqrt{\frac{\sum{(x_i - \bar{x})^2 \cdot f_i}}{\sum{f_i}}} \][/tex]
Breaking it down:
[tex]\[ \sigma = \sqrt{\frac{\sum{f \cdot (x - \bar{x})^2}}{\sum{f}}} \][/tex]
Calculating each term:
[tex]\[ (500 - 846.5)^2 \times 7 = 85312.25 \times 7 \][/tex]
[tex]\[ (600 - 846.5)^2 \times 20 = 61022.5 \times 20 \][/tex]
[tex]\[ (700 - 846.5)^2 \times 26 = 21522.5 \times 26 \][/tex]
[tex]\[ (800 - 846.5)^2 \times 47 = 2162.25 \times 47 \][/tex]
[tex]\[ (900 - 846.5)^2 \times 50 = 2852.25 \times 50 \][/tex]
[tex]\[ (1000 - 846.5)^2 \times 34 = 23552.25 \times 34 \][/tex]
[tex]\[ (1100 - 846.5)^2 \times 12 = 64352.25 \times 12 \][/tex]
[tex]\[ (1200 - 846.5)^2 \times 4 = 125702.25 \times 4 \][/tex]
[tex]\[ = 597178 + 1220450 + 559612 + 101623 + 142612 + 800678 + 772227 + 502809 \][/tex]
Summing these values, we get approximately the total sum of squared deviations:
[tex]\[ = 4824189 \][/tex]
Finally:
[tex]\[ \sigma = \sqrt{\frac{4824189}{200}} \][/tex]
[tex]\[ \sigma \approx 156.93231024871838 \][/tex]
So, the standard deviation of the data is approximately 156.93231024871838 kWh.
We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.