Answer :
To determine in which year the actual population of Center City was most different from the value predicted by the linear model, we need to follow these steps:
1. Construct a Linear Model Using Given Data:
- We are given the populations for the years 1990 and 2005:
[tex]\[ \text{Year 1990, Population: } 197,800 \][/tex]
[tex]\[ \text{Year 2005, Population: } 206,561 \][/tex]
- To find the linear model of the form [tex]\(y = mx + b\)[/tex], we first need to calculate the slope ([tex]\(m\)[/tex]). The slope is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the values, we get:
[tex]\[ m = \frac{206,561 - 197,800}{2005 - 1990} = \frac{8,761}{15} = 584.0667 \][/tex]
- Next, we calculate the intercept ([tex]\(b\)[/tex]) using the slope and one data point [tex]\((x_1, y_1)\)[/tex]:
[tex]\[ b = y_1 - mx_1 \][/tex]
Using the point (1990, 197,800):
[tex]\[ b = 197,800 - 584.0667 \times 1990 = 197,800 - 1,162,292.6667 = -964,492.6667 \][/tex]
- Now, we have the linear model:
[tex]\[ y = 584.0667x - 964,492.6667 \][/tex]
2. Predict the Population for the Given Years:
- Using the linear model, we calculate the predicted population for the years 1985, 1992, 2000, and 2012:
[tex]\[ \begin{align*} \text{For } 1985: & \quad y = 584.0667 \times 1985 - 964,492.6667 \approx 194,879.67 \\ \text{For } 1992: & \quad y = 584.0667 \times 1992 - 964,492.6667 \approx 198,968.13 \\ \text{For } 2000: & \quad y = 584.0667 \times 2000 - 964,492.6667 \approx 203,640.67 \\ \text{For } 2012: & \quad y = 584.0667 \times 2012 - 964,492.6667 \approx 210,649.47 \\ \end{align*} \][/tex]
3. Calculate the Differences Between Actual and Predicted Populations:
- The actual populations for the years are given in the table. We compute the absolute differences:
[tex]\[ \begin{align*} \text{For } 1985: & \quad |194,957 - 194,879.67| \approx 77.33 \\ \text{For } 1992: & \quad |199,532 - 198,968.13| \approx 563.87 \\ \text{For } 2000: & \quad |203,750 - 203,640.67| \approx 109.33 \\ \text{For } 2012: & \quad |210,600 - 210,649.47| \approx 49.47 \\ \end{align*} \][/tex]
4. Identify the Year with Maximum Difference:
- Comparing the differences:
[tex]\[ \begin{align*} 1985 & : 77.33 \\ 1992 & : 563.87 \\ 2000 & : 109.33 \\ 2012 & : 49.47 \\ \end{align*} \][/tex]
- The year with the maximum difference between actual and predicted populations is 1992, with a difference of approximately 563.87.
Therefore, the actual population of Center City was most different from the value predicted by the linear model in the year 1992.
1. Construct a Linear Model Using Given Data:
- We are given the populations for the years 1990 and 2005:
[tex]\[ \text{Year 1990, Population: } 197,800 \][/tex]
[tex]\[ \text{Year 2005, Population: } 206,561 \][/tex]
- To find the linear model of the form [tex]\(y = mx + b\)[/tex], we first need to calculate the slope ([tex]\(m\)[/tex]). The slope is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the values, we get:
[tex]\[ m = \frac{206,561 - 197,800}{2005 - 1990} = \frac{8,761}{15} = 584.0667 \][/tex]
- Next, we calculate the intercept ([tex]\(b\)[/tex]) using the slope and one data point [tex]\((x_1, y_1)\)[/tex]:
[tex]\[ b = y_1 - mx_1 \][/tex]
Using the point (1990, 197,800):
[tex]\[ b = 197,800 - 584.0667 \times 1990 = 197,800 - 1,162,292.6667 = -964,492.6667 \][/tex]
- Now, we have the linear model:
[tex]\[ y = 584.0667x - 964,492.6667 \][/tex]
2. Predict the Population for the Given Years:
- Using the linear model, we calculate the predicted population for the years 1985, 1992, 2000, and 2012:
[tex]\[ \begin{align*} \text{For } 1985: & \quad y = 584.0667 \times 1985 - 964,492.6667 \approx 194,879.67 \\ \text{For } 1992: & \quad y = 584.0667 \times 1992 - 964,492.6667 \approx 198,968.13 \\ \text{For } 2000: & \quad y = 584.0667 \times 2000 - 964,492.6667 \approx 203,640.67 \\ \text{For } 2012: & \quad y = 584.0667 \times 2012 - 964,492.6667 \approx 210,649.47 \\ \end{align*} \][/tex]
3. Calculate the Differences Between Actual and Predicted Populations:
- The actual populations for the years are given in the table. We compute the absolute differences:
[tex]\[ \begin{align*} \text{For } 1985: & \quad |194,957 - 194,879.67| \approx 77.33 \\ \text{For } 1992: & \quad |199,532 - 198,968.13| \approx 563.87 \\ \text{For } 2000: & \quad |203,750 - 203,640.67| \approx 109.33 \\ \text{For } 2012: & \quad |210,600 - 210,649.47| \approx 49.47 \\ \end{align*} \][/tex]
4. Identify the Year with Maximum Difference:
- Comparing the differences:
[tex]\[ \begin{align*} 1985 & : 77.33 \\ 1992 & : 563.87 \\ 2000 & : 109.33 \\ 2012 & : 49.47 \\ \end{align*} \][/tex]
- The year with the maximum difference between actual and predicted populations is 1992, with a difference of approximately 563.87.
Therefore, the actual population of Center City was most different from the value predicted by the linear model in the year 1992.
