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.

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 \(\boxed{0.256}\).