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Chapter 1: Problem 57

At the beginning of the 2007 baseball season each American League team had nine starting position players (this includes the designated hitter but not the pitcher). Here are the salaries for the New York Yankees and the Cleveland Indians in thousands of dollars: \(\begin{array}{llllll}\text { Yankees: } & 12000 & 600 & 491 & 22709 & 21600 \\\ & 13000 & 13000 & 15000 & 23429 & \\ \text { Indians: } & 3200 & 3750 & 396 & 383 & 1000 \\ & 3750 & 917 & 3000 & 4050 & \end{array}\) Construct a comparative boxplot and comment on interesting features. Compare the salaries of the two teams. The Indians won more games than the Yankees in the regular season and defeated the Yankees in the playoffs.

Short Answer

Expert verified
Indians' salaries are lower but they performed better than Yankees. Higher salaries do not guarantee better performance.

Step by step solution

01

Organize the Data

First, list the salaries for each team in an ascending order. For the Yankees: 491, 600, 12000, 13000, 13000, 15000, 21600, 22709, 23429. For the Indians: 383, 396, 917, 1000, 3000, 3200, 3750, 3750, 4050.
02

Find the Five-Number Summary

Calculate the minimum, first quartile (Q1), median, third quartile (Q3), and maximum for each team. For the Yankees: - Minimum = 491 - Q1 = 600 - Median = 13000 - Q3 = 21600 - Maximum = 23429 For the Indians: - Minimum = 383 - Q1 = 917 - Median = 3000 - Q3 = 3750 - Maximum = 4050.
03

Construct Boxplots

Using the five-number summaries, construct boxplots for both teams on the same scale to facilitate comparison. - Each boxplot should have lines at the minimum, Q1, median, Q3, and maximum. - The box for the Yankees will be much longer due to higher salary differences, especially in the Q3 to maximum range.
04

Compare the Boxplots

Analyze the boxplots visually and statistically. - The Yankees have a higher range of salaries with a larger spread between Q1 and Q3. - The Yankees' upper quartile shows extremely high salaries (outliers possible). - The Indians' boxplot is more compressed, indicating closer salary values within the team.
05

Comment on Findings

Comment on the surprising fact regarding performance versus salary. - Despite higher salaries, the Yankees did not outperform the Indians in terms of games won or playoff success. - This suggests that higher salary does not necessarily equate to better team performance.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Five-Number Summary
Understanding a five-number summary is crucial for data analysis, especially when comparing datasets. A five-number summary consists of:
  • Minimum: The smallest data point.
  • First Quartile (Q1): The median of the lower half of the data.
  • Median: The middle value of the dataset.
  • Third Quartile (Q3): The median of the upper half of the data.
  • Maximum: The largest data point.
The purpose of this summary is to provide a quick snapshot of the dataset's distribution.

In the provided problem, the five-number summaries were calculated for the Yankees and Indians. This helps in understanding their salary distributions.

The Yankees' salary distribution ranged from 491 to 23,429 with the median at 13,000, indicating a high salary spread. The Indians' salaries varied from 383 to 4,050 with a median of 3,000, showing tighter clustering with lower maximum values. This summary helps in identifying the characteristics and spread of the salaries of each team.
Data Analysis
Data analysis involves organizing and interpreting data to extract meaningful insights. In this exercise, data analysis starts with organizing the salaries in ascending order. This allows easier calculation of key statistics like the five-number summary.

The organization step aids in spotting trends, such as the Yankees having much higher salaries compared to the Indians.

When analyzing the data, it's important to discern patterns and deviations. For instance, the Yankees show larger variability, while the Indians exhibit more consistent salaries.
  • Organized Data: Essential for accurate calculations of medians and quartiles.
  • Pattern Recognition: Identifying spread, clustering and outliers can lead to insights about the data set.
This deeper look reveals that high variance within the Yankees' salaries might contribute to employment of higher-paid outliers.
Comparative Statistics
Comparative statistics allow us to compare datasets like the Yankees' and Indians' salaries by examining key statistics side by side. The construction of boxplots based on the five-number summaries is a primary method for comparison.

Boxplots visually represent the differences in salary distributions. They highlight similarities and differences, helping to compare central tendencies and spreads between datasets.
  • Visual Insights: Boxplots make it easy to see salary spread and identify any potential outliers.
  • Statistical Details: The larger range and median suggest that the Yankees pay higher salaries overall compared to the Indians.

Such comparative analysis aids in understanding the counterintuitive outcome that higher salaries did not translate to better performance, as seen with the Indians outperforming the Yankees. This comparison showcases that salary distribution can impact team dynamics beyond just a simple high-on-paper salary might imply.

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Most popular questions from this chapter

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