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To analyze the provided data for User Fees and Operating Costs, we will perform the following steps:
1. Calculate the Mean of User Fees:
The mean (average) is calculated by summing all the values in the User Fees dataset and then dividing by the number of values.
User Fees: [tex]\(17, 74, 811, 380, 33, 427, 500, 734, 760\)[/tex]
Step-by-step calculation:
[tex]\[ \text{Sum of User Fees} = 17 + 74 + 811 + 380 + 33 + 427 + 500 + 734 + 760 = 3,736 \][/tex]
Number of User Fees values = 9
Mean of User Fees:
[tex]\[ \text{Mean of User Fees} = \frac{3,736}{9} \approx 415.11 \text{ (thousands of dollars)} \][/tex]
2. Calculate the Mean of Operating Costs:
Similarly, we calculate the mean of the Operating Costs dataset by summing all the values and then dividing by the number of values.
Operating Costs: [tex]\(101, 355, 3,546, 1,363, 243, 768, 1,054, 2,227, 1,604\)[/tex]
Step-by-step calculation:
[tex]\[ \text{Sum of Operating Costs} = 101 + 355 + 3,546 + 1,363 + 243 + 768 + 1,054 + 2,227 + 1,604 = 11,261 \][/tex]
Number of Operating Costs values = 9
Mean of Operating Costs:
[tex]\[ \text{Mean of Operating Costs} = \frac{11,261}{9} \approx 1251.22 \text{ (thousands of dollars)} \][/tex]
3. Calculate the Correlation between User Fees and Operating Costs:
Correlation measures the relationship between two datasets. Here, we want to find the correlation between User Fees and Operating Costs. The correlation coefficient, denoted as [tex]\( r \)[/tex], will lie between -1 and 1, where 1 indicates a perfect positive correlation, -1 a perfect negative correlation, and 0 no correlation.
The formula for the Pearson correlation coefficient [tex]\( r \)[/tex] is:
[tex]\[ r = \frac{n \sum (XY) - \sum X \sum Y}{\sqrt{[n \sum X^2 - (\sum X)^2][n \sum Y^2 - (\sum Y)^2]}} \][/tex]
Here:
- [tex]\( X \)[/tex] represents the User Fees
- [tex]\( Y \)[/tex] represents the Operating Costs
- [tex]\( n \)[/tex] is the number of pairs (which is 9)
Given the correlation between User Fees and Operating Costs is:
[tex]\[ r \approx 0.878 \][/tex]
This value indicates a strong positive correlation between User Fees and Operating Costs.
### Summary:
- Mean of User Fees: Approximately [tex]\( 415.11 \)[/tex] thousand dollars.
- Mean of Operating Costs: Approximately [tex]\( 1251.22 \)[/tex] thousand dollars.
- Correlation between User Fees and Operating Costs: Approximately [tex]\( 0.878 \)[/tex], indicating a strong positive correlation.
1. Calculate the Mean of User Fees:
The mean (average) is calculated by summing all the values in the User Fees dataset and then dividing by the number of values.
User Fees: [tex]\(17, 74, 811, 380, 33, 427, 500, 734, 760\)[/tex]
Step-by-step calculation:
[tex]\[ \text{Sum of User Fees} = 17 + 74 + 811 + 380 + 33 + 427 + 500 + 734 + 760 = 3,736 \][/tex]
Number of User Fees values = 9
Mean of User Fees:
[tex]\[ \text{Mean of User Fees} = \frac{3,736}{9} \approx 415.11 \text{ (thousands of dollars)} \][/tex]
2. Calculate the Mean of Operating Costs:
Similarly, we calculate the mean of the Operating Costs dataset by summing all the values and then dividing by the number of values.
Operating Costs: [tex]\(101, 355, 3,546, 1,363, 243, 768, 1,054, 2,227, 1,604\)[/tex]
Step-by-step calculation:
[tex]\[ \text{Sum of Operating Costs} = 101 + 355 + 3,546 + 1,363 + 243 + 768 + 1,054 + 2,227 + 1,604 = 11,261 \][/tex]
Number of Operating Costs values = 9
Mean of Operating Costs:
[tex]\[ \text{Mean of Operating Costs} = \frac{11,261}{9} \approx 1251.22 \text{ (thousands of dollars)} \][/tex]
3. Calculate the Correlation between User Fees and Operating Costs:
Correlation measures the relationship between two datasets. Here, we want to find the correlation between User Fees and Operating Costs. The correlation coefficient, denoted as [tex]\( r \)[/tex], will lie between -1 and 1, where 1 indicates a perfect positive correlation, -1 a perfect negative correlation, and 0 no correlation.
The formula for the Pearson correlation coefficient [tex]\( r \)[/tex] is:
[tex]\[ r = \frac{n \sum (XY) - \sum X \sum Y}{\sqrt{[n \sum X^2 - (\sum X)^2][n \sum Y^2 - (\sum Y)^2]}} \][/tex]
Here:
- [tex]\( X \)[/tex] represents the User Fees
- [tex]\( Y \)[/tex] represents the Operating Costs
- [tex]\( n \)[/tex] is the number of pairs (which is 9)
Given the correlation between User Fees and Operating Costs is:
[tex]\[ r \approx 0.878 \][/tex]
This value indicates a strong positive correlation between User Fees and Operating Costs.
### Summary:
- Mean of User Fees: Approximately [tex]\( 415.11 \)[/tex] thousand dollars.
- Mean of Operating Costs: Approximately [tex]\( 1251.22 \)[/tex] thousand dollars.
- Correlation between User Fees and Operating Costs: Approximately [tex]\( 0.878 \)[/tex], indicating a strong positive correlation.
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