To solve this problem, we need to find the line of best fit for the given data and then use it to predict the number of games the player would win if they played 40 games.
Step 1: Determine the Line of Best Fit
We have the following data points for games played (x) and games won (y):
[tex]\[
\begin{array}{|c|c|c|c|c|c|}
\hline
\text{Games Played} \, (x) & 10 & 15 & 20 & 25 & 30 \\
\hline
\text{Games Won} \, (y) & 4 & 10 & 16 & 21 & 25 \\
\hline
\end{array}
\][/tex]
Using these data points, the line of best fit can be obtained. This line can be represented in the linear equation form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Here, the slope [tex]\( m \)[/tex] and y-intercept [tex]\( b \)[/tex] are:
[tex]\[
m = 1.0600000000000003
\][/tex]
[tex]\[
b = -5.9999999999999964
\][/tex]
Thus, the line of best fit is:
[tex]\[
y = 1.06x - 6.00
\][/tex]
Step 2: Predict the Number of Games Won for 40 Games Played
To find out the expected number of games won when the chess player plays 40 games, we substitute [tex]\( x = 40 \)[/tex] into the line of best fit equation.
[tex]\[
y = 1.06 \times 40 - 6.00
\][/tex]
Performing the multiplication and subtraction, we get:
[tex]\[
y = 42.4 - 6.00 = 36.4
\][/tex]
Thus, the expected number of games won for 40 games played is approximately 36.4.
So, the correct choices from the drop-down menu are:
- The line of best fit for the situation is [tex]\( y = 1.06x - 6.00 \)[/tex].
- If the chess player plays 40 games in the next competition, the expected number of games won would be approximately 36.4.
