To find the line of best fit and the expected number of games won when the chess player plays 40 games, follow these steps:
### Step 1: Construct the Data Points
You have the following data points from the table:
- Games Played [tex]$(x)$[/tex]: 10, 15, 20, 25, 30
- Games Won [tex]$(y)$[/tex]: 4, 10, 16, 21, 25
### Step 2: Perform Linear Regression
To determine the line of best fit, a linear regression analysis is conducted, which finds the relationship in the form \( y = mx + c \). Here:
- \( m \) represents the slope of the line.
- \( c \) represents the y-intercept.
The result from the analysis gives us:
- \( m = 1.06 \)
- \( c = -6.00 \)
So, the equation of the line of best fit is:
[tex]\[ y = 1.06x - 6.00 \][/tex]
### Step 3: Predict the Number of Games Won for 40 Games Played
To find the expected number of games won when the player plays 40 games, substitute \( x = 40 \) into the equation of the line of best fit:
[tex]\[ y = 1.06(40) - 6.00 \][/tex]
Calculating this:
[tex]\[ y = 42.4 - 6.00 \][/tex]
[tex]\[ y = 36.4 \][/tex]
Thus, if the chess player plays 40 games in the next competition, the expected number of games won would be approximately 36.40.
### Final Answer:
The line of best fit for the situation is \( y = 1.06x - 6.00 \).
If the chess player plays 40 games in the next competition, the expected number of games won would be approximately 36.4.
