To determine which quadratic function best fits the given data, we examine the sum of squares of residuals (SSR) for each candidate quadratic function. The residuals are the differences between the observed values and the values predicted by the model. The SSR is computed as follows:
[tex]\[ \text{SSR} = \sum (y_{\text{predicted}} - y_{\text{observed}})^2 \][/tex]
For the given data:
[tex]\[
\begin{array}{|c|c|}
\hline
x & y \\
\hline
1 & 250 \\
2 & 289 \\
3 & 316 \\
4 & 335 \\
5 & 320 \\
6 & 290 \\
\hline
\end{array}
\][/tex]
We have four candidate quadratic functions:
1. [tex]\( y = 9.16 x^2 - 73.04 x + 183.3 \)[/tex]
2. [tex]\( y = 9.16 x^2 + 73.04 x + 183.3 \)[/tex]
3. [tex]\( y = -9.16 x^2 + 73.04 x + 183.3 \)[/tex]
4. [tex]\( y = -9.16 x^2 - 73.04 x + 183.3 \)[/tex]
For each function, we calculate the SSR:
- For the first function [tex]\( y = 9.16 x^2 - 73.04 x + 183.3 \)[/tex], the SSR is [tex]\( 345066.0728 \)[/tex].
- For the second function [tex]\( y = 9.16 x^2 + 73.04 x + 183.3 \)[/tex], the SSR is [tex]\( 763690.4504 \)[/tex].
- For the third function [tex]\( y = -9.16 x^2 + 73.04 x + 183.3 \)[/tex], the SSR is [tex]\( 78.4087999999998 \)[/tex].
- For the fourth function [tex]\( y = -9.16 x^2 - 73.04 x + 183.3 \)[/tex], the SSR is [tex]\( 1941849.7304 \)[/tex].
The function with the smallest SSR is the one that best fits the data. From the values above, the smallest SSR is [tex]\( 78.4087999999998 \)[/tex], which corresponds to the third function [tex]\( y = -9.16 x^2 + 73.04 x + 183.3 \)[/tex].
Therefore, the quadratic function that best fits the given data is:
[tex]\[ y = -9.16 x^2 + 73.04 x + 183.3 \][/tex]
