To determine which function accurately fits the data given in the table, we need to evaluate each provided function against the data points and find the one with the smallest error. A common way to measure the fit is to calculate the Mean Squared Error (MSE) for each function. The function with the smallest MSE is considered the best fit.
For each hour [tex]\( x \)[/tex] and the corresponding number of customers [tex]\( y \)[/tex], we have the following data points:
[tex]\[
\begin{tabular}{|c|c|}
\hline
Hour (\( x \)) & Customers (\( y \)) \\
\hline
1 & 15 \\
\hline
2 & 26 \\
\hline
3 & 33 \\
\hline
4 & 36 \\
\hline
5 & 35 \\
\hline
6 & 30 \\
\hline
7 & 21 \\
\hline
\end{tabular}
\][/tex]
We test the following four functions:
1. [tex]\( y = 2x^2 - 17x \)[/tex]
2. [tex]\( y = -2x^2 + 17x \)[/tex]
3. [tex]\( y = x + 24 \)[/tex]
4. [tex]\( y = -2x + 17 \)[/tex]
Calculate the Mean Squared Error (MSE) for each function:
- For [tex]\( y = 2x^2 - 17x \)[/tex]: [tex]\( \text{MSE}_A = 3344.0 \)[/tex]
- For [tex]\( y = -2x^2 + 17x \)[/tex]: [tex]\( \text{MSE}_B = 0.0 \)[/tex]
- For [tex]\( y = x + 24 \)[/tex]: [tex]\( \text{MSE}_C = 48.0 \)[/tex]
- For [tex]\( y = -2x + 17 \)[/tex]: [tex]\( \text{MSE}_D = 445.0 \)[/tex]
Comparing the MSE values, we find that [tex]\( y = -2x^2 + 17x \)[/tex] has the smallest MSE of [tex]\( 0.0 \)[/tex]. Thus, this function provides the best fit for the data.
The correct answer is:
b. B
