Baseball regression line prediction:
Suppose the regression line for the number of runs scored in a season, $y$, is given by
\[
\hat{y}=-800+5000 x
\]
where $x$ is the team's batting average.
a. For a team with a batting average of 0.235 , find the expected number of runs scored in a season. Round your answer to the nearest whole number.
b. If we can expect the number of runs scored in a season is 480 , then what is the assumed team's batting average? Round your answer to three decimal places.
So, if we can expect the number of runs scored in a season is 480, then the assumed team's batting average is \(\boxed{0.256}\).
Step 1 :Given the regression line for the number of runs scored in a season, \(\hat{y}=-800+5000 x\), where \(x\) is the team's batting average.
Step 2 :For part a, we need to substitute the given batting average into the equation and solve for \(\hat{y}\), which represents the expected number of runs. Let's substitute \(x = 0.235\) into the equation.
Step 3 :\(\hat{y}=-800+5000 \times 0.235\)
Step 4 :\(\hat{y}=375\)
Step 5 :So, the expected number of runs scored in a season for a team with a batting average of 0.235 is \(\boxed{375}\).
Step 6 :For part b, we need to substitute the given number of runs into the equation and solve for \(x\), which represents the team's batting average. Let's substitute \(\hat{y} = 480\) into the equation.
Step 7 :\(480=-800+5000 x\)
Step 8 :Solving for \(x\), we get \(x = 0.256\)
Step 9 :So, if we can expect the number of runs scored in a season is 480, then the assumed team's batting average is \(\boxed{0.256}\).