Problem 39 - An Introduction to Mathematical Statistics and Its Applications

Question

A utility infielder for a National League club batted . 260 last season in three hundred trips to the plate. This year he hit \(.250\) in two hundred at- bats. The owners are trying to cut his pay for next year on the grounds that his output has deteriorated. The player argues, though, that his performances the last two seasons have not been significantly different, so his salary should not be reduced. Who is right?

Step-by-Step Solution

Answer

The player is right. His overall performance over the two years has not decreased significantly from his first year's performance. Therefore, reducing his salary may not be justified.

Step 1: Calculate Number of Hits for Each Year

For the first year, take the batting average of \(.260\) and multiply it by 300 at-bats, which is a total of 78 hits. For the second year, multiply the \(.250\) batting average by 200 at-bats, which equals to 50 hits.

Step 2: Calculate Total Batting Average Over Both Years

To get the overall batting average over these two years, sum up the total hits (78 + 50 = 128 hits) and divide it by the total number of at-bats (300 + 200 = 500 at-bats). This results in an average of \(.256\).

Step 3: Analyze the Result

The batting average of the player is \(.256\) over the two years. This is not significantly less than the first year's average (\(.260\)), but higher than his second year's average (\(.250\)).

Key Concepts

Batting Average

The batting average is a crucial statistic for evaluating a baseball player's performance. It signifies how often a player gets a hit during their at-bats. To calculate the batting average, you divide the number of hits by the number of times the player has batted. This is represented in the formula:

• Batting Average = \( \frac{\text{Number of Hits}}{\text{Number of At-Bats}} \)

For example, if a player had 100 hits in 400 at-bats, their batting average would be \(0.250\). This means, on average, the player got a hit in 25% of their at-bats.
Moreover, a batting average is often represented as a decimal and noted to three decimal places. This provides a detailed insight into the player's hitting efficiency over the season. By focusing on this single metric, teams can quickly analyze individual performance throughout the season.

Performance Comparison

Comparing player performance across different seasons or contexts relies heavily on consistent metrics like the batting average. In our example, the utility infielder's performance was compared over two seasons.
To determine if there is a significant change, you assess each season's batting average. The player had a \(0.260\) average in one season and \(0.250\) in another.

• In the first year, the player demonstrated better performance with a \(0.260\) average.
• In the second year, a slight decrease to \(0.250\) was observed.
• Overall two-year average at \(0.256\), reflects minor change in performance.

Such comparisons help in understanding whether changes in performance are drastic or minor, assisting in evaluations related to player contracts or salary negotiations.

Sports Statistics

Sports statistics provide essential insights into the performance and potential of athletes. Statistics like the batting average help owners, coaches, and players make informed decisions. They allow for objective comparisons and assessments.

• Sports stats can reveal trends such as improvement or decline over time.
• They offer a numerical value to skills otherwise open to subjective interpretation.
• These statistics help in predicting future performance and making data-driven decisions.

In competitive sports, relying solely on observation isn't enough. Numerical metrics give a tangible measure of performance, aiding contract discussions and strategy planning. Statistics are an indispensable tool in the business side of sports, affecting managerial and salary decisions based on factual data rather than perception.