Find the regression line for the following data. Round values to the nearest tenth if necessary.

\begin{tabular}{|l|l|}
\hline
Day [tex]$(x)$[/tex] & Number of visitors [tex]$(y)$[/tex] \\
\hline
1 & 120 \\
\hline
2 & 124 \\
\hline
3 & 130 \\
\hline
4 & 131 \\
\hline
5 & 135 \\
\hline
6 & 132 \\
\hline
7 & 135 \\
\hline
\end{tabular}

A. [tex]$y = 0.3x - 41.1$[/tex]
B. [tex]$y = 2.4x + 120.1$[/tex]
C. [tex]$y = 0.3x - 29$[/tex]
D. [tex]$y = 4x + 116$[/tex]



Answer :

To determine the regression line for the given data, we'll follow these steps.

1. Collect the Data:
We have the following pairs of [tex]\( x \)[/tex] (Day) and [tex]\( y \)[/tex] (Number of visitors):
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 1 & 120 \\ \hline 2 & 124 \\ \hline 3 & 130 \\ \hline 4 & 131 \\ \hline 5 & 135 \\ \hline 6 & 132 \\ \hline 7 & 135 \\ \hline \end{array} \][/tex]

2. Fit a Linear Regression Line:
A linear regression line can be expressed in the form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept. Using statistical methods, we can determine these coefficients.

3. Interpret the Results:
After performing the calculations, we get the following coefficients:
- Slope ([tex]\( m \)[/tex]): [tex]\( 2.4 \)[/tex]
- Intercept ([tex]\( b \)[/tex]): [tex]\( 120.1 \)[/tex]

4. Express the Regression Line:
With [tex]\( m = 2.4 \)[/tex] and [tex]\( b = 120.1 \)[/tex], the equation of the regression line is:
[tex]\[ y = 2.4x + 120.1 \][/tex]

5. Choose the Correct Option:
Comparing our result with the given choices:
- A: [tex]\( y = 0.3x - 41.1 \)[/tex]
- B: [tex]\( y = 2.4x + 120.1 \)[/tex]
- C: [tex]\( y = 0.3x - 29 \)[/tex]
- D: [tex]\( y = 4x + 116 \)[/tex]

The correct option that matches our regression line equation is:
[tex]\[ \boxed{B: \, y = 2.4x + 120.1} \][/tex]