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To create a linear model that predicts the population [tex]\( y \)[/tex] of Center City in a given year [tex]\( x \)[/tex], we will use the data from the years 1990 and 2005. Let's break this down step-by-step:
1. Identify the given data for the years 1990 and 2005:
- Year 1990: Population is 197,800
- Year 2005: Population is 206,561
2. Determine the slope [tex]\( m \)[/tex] of the linear model:
The slope of the line connecting the points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be calculated using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the values:
[tex]\[ m = \frac{206,561 - 197,800}{2005 - 1990} = \frac{8,761}{15} \approx 584.0667 \][/tex]
3. Find the y-intercept [tex]\( b \)[/tex] of the linear model:
To find the y-intercept, use the point-slope form of the line equation:
[tex]\[ b = y_1 - m \cdot x_1 \][/tex]
Substituting the values from the year 1990:
[tex]\[ b = 197,800 - 584.0667 \cdot 1990 \approx 197,800 - 1,162,292.6667 \approx -964,492.6667 \][/tex]
4. Write the linear equation:
The linear model predicting the population [tex]\( y \)[/tex] in a given year [tex]\( x \)[/tex] is:
[tex]\[ y = 584.0667 \cdot x - 964,492.6667 \][/tex]
5. Predict the populations using the linear model for the given years:
- 1985: [tex]\( y = 584.0667 \cdot 1985 - 964,492.6667 \approx 194,879.67 \)[/tex]
- 1990: [tex]\( y = 197,800 \)[/tex]
- 1992: [tex]\( y = 584.0667 \cdot 1992 - 964,492.6667 \approx 198,968.13 \)[/tex]
- 2000: [tex]\( y = 584.0667 \cdot 2000 - 964,492.6667 \approx 203,640.67 \)[/tex]
- 2005: [tex]\( y = 206,561 \)[/tex]
- 2012: [tex]\( y = 584.0667 \cdot 2012 - 964,492.6667 \approx 210,649.47 \)[/tex]
6. Calculate the differences between the actual and predicted populations:
- 1985: [tex]\( \left| 194,957 - 194,879.67 \right| \approx 77.33 \)[/tex]
- 1990: [tex]\( \left| 197,800 - 197,800 \right| = 0 \)[/tex]
- 1992: [tex]\( \left| 199,532 - 198,968.13 \right| \approx 563.87 \)[/tex]
- 2000: [tex]\( \left| 203,750 - 203,640.67 \right| \approx 109.33 \)[/tex]
- 2005: [tex]\( \left| 206,561 - 206,561 \right| = 0 \)[/tex]
- 2012: [tex]\( \left| 210,600 - 210,649.47 \right| \approx 49.47 \)[/tex]
7. Identify the year with the maximum difference between the actual and predicted population:
By examining the differences, we see the maximum difference occurs in the year 1992, with a difference of approximately 563.87.
Therefore, the actual population of Center City was most different from the value predicted by the model in the year 1992.
1. Identify the given data for the years 1990 and 2005:
- Year 1990: Population is 197,800
- Year 2005: Population is 206,561
2. Determine the slope [tex]\( m \)[/tex] of the linear model:
The slope of the line connecting the points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] can be calculated using the formula:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the values:
[tex]\[ m = \frac{206,561 - 197,800}{2005 - 1990} = \frac{8,761}{15} \approx 584.0667 \][/tex]
3. Find the y-intercept [tex]\( b \)[/tex] of the linear model:
To find the y-intercept, use the point-slope form of the line equation:
[tex]\[ b = y_1 - m \cdot x_1 \][/tex]
Substituting the values from the year 1990:
[tex]\[ b = 197,800 - 584.0667 \cdot 1990 \approx 197,800 - 1,162,292.6667 \approx -964,492.6667 \][/tex]
4. Write the linear equation:
The linear model predicting the population [tex]\( y \)[/tex] in a given year [tex]\( x \)[/tex] is:
[tex]\[ y = 584.0667 \cdot x - 964,492.6667 \][/tex]
5. Predict the populations using the linear model for the given years:
- 1985: [tex]\( y = 584.0667 \cdot 1985 - 964,492.6667 \approx 194,879.67 \)[/tex]
- 1990: [tex]\( y = 197,800 \)[/tex]
- 1992: [tex]\( y = 584.0667 \cdot 1992 - 964,492.6667 \approx 198,968.13 \)[/tex]
- 2000: [tex]\( y = 584.0667 \cdot 2000 - 964,492.6667 \approx 203,640.67 \)[/tex]
- 2005: [tex]\( y = 206,561 \)[/tex]
- 2012: [tex]\( y = 584.0667 \cdot 2012 - 964,492.6667 \approx 210,649.47 \)[/tex]
6. Calculate the differences between the actual and predicted populations:
- 1985: [tex]\( \left| 194,957 - 194,879.67 \right| \approx 77.33 \)[/tex]
- 1990: [tex]\( \left| 197,800 - 197,800 \right| = 0 \)[/tex]
- 1992: [tex]\( \left| 199,532 - 198,968.13 \right| \approx 563.87 \)[/tex]
- 2000: [tex]\( \left| 203,750 - 203,640.67 \right| \approx 109.33 \)[/tex]
- 2005: [tex]\( \left| 206,561 - 206,561 \right| = 0 \)[/tex]
- 2012: [tex]\( \left| 210,600 - 210,649.47 \right| \approx 49.47 \)[/tex]
7. Identify the year with the maximum difference between the actual and predicted population:
By examining the differences, we see the maximum difference occurs in the year 1992, with a difference of approximately 563.87.
Therefore, the actual population of Center City was most different from the value predicted by the model in the year 1992.
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