2. The highest batting average ever recorded in Major League Baseball was by Ted
Williams in 1941 when he hit 0.406. That year, the mean and standard deviation for batting
average were 0.2806 and 0.0328. In 2014, Jose Altuve was the American League batting
champion, with a batting average of 0.341. In 2014, the mean and standard deviation for
batting average were 0.2679 and 0.0282. Who had the better year relative to his peers,
Williams or Altuve? Why?

Ted Williams had a better year relative to his peers because the z-score and percentage of his batting average in 1941 are higher than Jose Altuve's z-score and percentage in 2014.

Step-by-step explanation:

1. Let's review the information given to us to answer the question correctly:

Ted Williams batting average in 1941 = 0.406

Mean of batting average in 1941 = 0.2806

Standard deviation of batting average in 1941 = 0.0328

Jose Altuve batting average in 2014 = 0.341

Mean of batting average in 1941 = 0.2679

Standard deviation of batting average in 1941 = 0.0282

2. Who had the better year relative to his peers, Williams or Altuve? Why?

Assuming a normal distribution for the batting average in Major League Baseball, we need to calculate the z-score to find out who had the better year relative to his peers, this way:

z-score Ted Williams = (0.406 - 0.2806)/0.0328

z-score Ted Williams = 0.1254/0.0328

z-score Ted Williams = 3.82

P for this z-score = 0.99993

z-score Jose Altuve = (0.341 - 0.2679)/0.0282

z-score Ted Williams = 0.0731/0.0282

z-score Ted Williams = 2.59

P for this z-score = 0.99520

3.82 > 2.59

Ted Williams had a better year relative to his peers because the z-score and percentage of his batting average in 1941 are higher than Jose Altuve's z-score and percentage in 2014.