Question

number of students (n) | measured points | predicted points | residual (measured-predicted)
4 | 90 | 89 | 1
8 | 350 | 337 | 13
12 | 750 | 745 | 5
16 | 1350 | 1313 | 37
20 | 2050 | |

the residuals show that the model the points earned based on the given data.

Explanation:

Identify the quadratic model

Using the Quadratic Modeling knowledge point
We need to find the quadratic model \(P(n) = an^2 + bn + c\) that relates the number of students \(n\) to the predicted points \(P(n)\).
Using the given predicted points:
For \(n = 12\), \(P(12) = 745\)
For \(n = 16\), \(P(16) = 1313\)
For \(n = 4\), \(P(4) = 89\)

Let's set up the system of equations:

1. \(16a + 4b + c = 89\)
2. \(144a + 12b + c = 745\)
3. \(256a + 16b + c = 1313\)

Subtracting (1) from (2):
\(128a + 8b = 656 \implies 16a + b = 82\)

Subtracting (2) from (3):
\(112a + 4b = 568 \implies 28a + b = 142\)

Subtracting these two simplified equations:
\(12a = 60 \implies a = 5\)

Substitute \(a = 5\) back:
\(16(5) + b = 82 \implies 80 + b = 82 \implies b = 2\)

Substitute \(a = 5\) and \(b = 2\) into (1):
\(16(5) + 4(2) + c = 89 \implies 80 + 8 + c = 89 \implies c = 1\)

Thus, the quadratic model is:

$$P(n) = 5n^2 + 2n + 1$$

Let's verify for \(n = 8\):

$$P(8) = 5(8^2) + 2(8) + 1 = 5(64) + 16 + 1 = 320 + 17 = 337$$

This matches the table value of 337.

Calculate predicted points for n = 20

Using the Quadratic Modeling knowledge point
We calculate the predicted points for \(n = 20\) using our model:

$$P(20) = 5(20^2) + 2(20) + 1$$

$$P(20) = 5(400) + 40 + 1 = 2000 + 41 = 2041$$

Calculate residual for n = 20

Using the Residual Calculation knowledge point
The residual is defined as:

$$\text{Residual} = \text{Measured} - \text{Predicted}$$

For \(n = 20\):

$$\text{Residual} = 2050 - 2041 = 9$$

Analyze the model performance

Using the Residual Calculation knowledge point
We examine the sign of all the residuals in the table:

• For \(n = 4\): \(\text{Residual} = 1 > 0\)
• For \(n = 8\): \(\text{Residual} = 13 > 0\)
• For \(n = 12\): \(\text{Residual} = 5 > 0\)
• For \(n = 16\): \(\text{Residual} = 37 > 0\)
• For \(n = 20\): \(\text{Residual} = 9 > 0\)

Since all residuals are positive, the measured values are consistently greater than the predicted values. This means the model consistently underpredicts (underestimates) the points earned.

Answer:

• Predicted Points for \(n = 20\): 2041
• Residual for \(n = 20\): 9
• Model evaluation statement: The residuals show that the model underestimates the points earned based on the given data.