Comparing batting averages Three landmarks of baseball achievement are Ty Cobb’s batting average of $420$in $1911$, Ted Williams’s $406$in $1941$, and George Brett’s $390$in $1980$. These batting averages cannot be compared directly because the distribution of major league batting averages has changed over the years. The distributions are quite symmetric, except for outliers such as Cobb, Williams, and Brett. While the mean batting average has been held roughly constant

by rule changes and the balance between hitting and pitching, the standard deviation has dropped over time. Here are the facts: Compute the standardized batting averages for Cobb, Williams, and Brett to compare how far each stood above his peers.

Step by step solution

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Step 1. Given

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Step 2. Concept

The inflection points are the points where the curve shifts from being concave up to being concave down. These inflection points are always one standard deviation distant from the mean on a normal density curve.

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Step 3. Calculation

Cobb's batting average in $1911$was $0.420$Cobb's standardized score can be found below.

$$\begin{gathered} z\left(Cobb\right)=x-\mu /\sigma \\ =0.420-0.266/0.0371 \\ \approx 4.15 \end{gathered}$$

In $1941$, William's batting average was $0.406$. The following is William's standardized score.

$$\begin{gathered} z\left(William\right)=x-\mu /\sigma \\ =0.406-0.267/0.0326 \\ \approx 4.26 \end{gathered}$$

In $1980$, Brett's batting average was $0.390$. Brett's standardized score may be found below.

$$\begin{gathered} z\left(Brett\right)=x-\mu /\sigma \\ =0.390-0.261/0.0317 \\ \approx 4.07 \end{gathered}$$

All three batters were at least four standard deviations above their contemporaries, but Williams had the highest Z-score. As a result, all three batters were at least four standard deviations ahead of their contemporaries, with W's Z-score being the highest.

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