Question

three landmarks of baseball achievement are ty cobbs batting average of 0.420 in 1911, ted williamss 0.406 in 1941, and george bretts 0.390 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:
decade mean standard deviation
1910s 0.266 0.0371
1940s 0.267 0.0326
1970s 0.261 0.0317
find the standardized scores for cobb, williams, and brett. who had the best performance for the decade he played?
cobb = 0.420
williams = 0.406
brett = 0.390
cobbs score is the highest.
cobb: z = 4.15
williams: z = 4.26
brett: z = 4.07
williams z - score is the highest.
cobb: z = - 6.75
williams: z = - 7.78
brett: z = - 7.84
cobbs z - score is the highest.
cobb: z = - 4.15
williams: z = - 4.26
brett: z = - 4.07
bretts z - score is the highest.

Explanation:

Step1: Recall the z - score formula

The formula for the z - score is $z=\frac{x - \mu}{\sigma}$, where $x$ is the data point, $\mu$ is the mean, and $\sigma$ is the standard deviation.

Step2: Calculate Cobb's z - score

Cobb played in 1911 (1910s decade). $\mu = 0.266$, $\sigma=0.0371$, and $x = 0.420$. Then $z_{Cobb}=\frac{0.420 - 0.266}{0.0371}=\frac{0.154}{0.0371}\approx4.15$.

Step3: Calculate Williams's z - score

Williams played in 1941 (1940s decade). $\mu = 0.267$, $\sigma = 0.0326$, and $x = 0.406$. Then $z_{Williams}=\frac{0.406- 0.267}{0.0326}=\frac{0.139}{0.0326}\approx4.26$.

Step4: Calculate Brett's z - score

Brett played in 1980 (1970s decade). $\mu = 0.261$, $\sigma = 0.0317$, and $x = 0.390$. Then $z_{Brett}=\frac{0.390 - 0.261}{0.0317}=\frac{0.129}{0.0317}\approx4.07$.

Step5: Compare the z - scores

Since $4.26>4.15>4.07$, Williams has the highest z - score.

Answer:

Cobb: $z = 4.15$
Williams: $z = 4.26$
Brett: $z = 4.07$
Williams' z - score is the highest.