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Chapter 3: Problem 10

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?

Short Answer

Expert verified
Ted Williams had the better year relative to his peers with a higher Z-score of approximately 3.82.

Step by step solution

01

Understand the Z-score Formula

The Z-score formula is used to determine how many standard deviations a data point is from the mean. The formula is: \[ Z = \frac{X - \text{mean}}{\text{standard deviation}} \] where \(X\) is the data point, the mean is the average, and the standard deviation measures the amount of variation from the mean.
02

Calculate Ted Williams' Z-score

For Ted Williams in 1941: \[ X = 0.406 \] \[ \text{mean} = 0.2806 \] \[ \text{standard deviation} = 0.0328 \] Substitute these values into the Z-score formula: \[ Z_{Williams} = \frac{0.406 - 0.2806}{0.0328} \approx 3.82 \]
03

Calculate Jose Altuve's Z-score

For Jose Altuve in 2014: \[ X = 0.341 \] \[ \text{mean} = 0.2679 \] \[ \text{standard deviation} = 0.0282 \] Substitute these values into the Z-score formula: \[ Z_{Altuve} = \frac{0.341 - 0.2679}{0.0282} \approx 2.59 \]
04

Compare the Z-scores

The Z-score represents how many standard deviations a data point is from the mean. A higher Z-score indicates a better performance relative to the peers. \[ Z_{Williams} \approx 3.82 \] \[ Z_{Altuve} \approx 2.59 \] Since Ted Williams' Z-score is higher than Jose Altuve's, Williams had the better year relative to his peers.

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Key Concepts

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

Batting Average
Batting average in baseball is a statistic that represents the performance of a batter. It's calculated by dividing the number of hits by the number of at bats. For example, if a player has 20 hits out of 50 at bats, their batting average would be 0.400. This means that the player hits the ball successfully 40% of the time. A higher batting average means better performance at the plate, as it indicates the batter is successfully hitting the ball more often. For reference, a batting average of 0.300 and above is generally considered excellent in Major League Baseball (MLB).
Standard Deviation
Standard deviation is a measure of how spread out the numbers in a data set are. In baseball, it tells us how much the batting averages deviate from the mean (average). A low standard deviation means that the data points (batting averages) are close to the mean, while a high standard deviation means they are spread out over a wider range of values.
As a statistical tool, standard deviation is essential for comparing individual performances within a group. For example, knowing the standard deviation of batting averages, we can better understand how exceptional or ordinary someone鈥檚 performance is compared to others.
Comparing Performance
When comparing performances, especially in sports like baseball, we don't just look at straightforward statistics like raw numbers. Instead, using tools like the Z-score helps us understand how a player鈥檚 performance stacks up against others under similar conditions. The Z-score calculates how many standard deviations a particular value (like a batting average) is from the mean of the dataset.
In this example, Ted Williams' batting average of 0.406 had a Z-score of approximately 3.82, indicating he was almost 4 standard deviations above the mean. Meanwhile, Jose Altuve鈥檚 0.341 batting average in 2014 had a Z-score of approximately 2.59, showing he was nearly 3 standard deviations above the mean.
  • A higher Z-score means a performance is better relative to the average.
  • It provides a way to compare across different years or leagues with different averages and deviations.
Since Williams had a higher Z-score, he had a better year relative to his peers compared to Altuve, even though their raw batting averages might suggest a simpler story.

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Most popular questions from this chapter

Ethan and Drew went on a 10 -day fishing trip. The number of smallmouth bass caught and released by the two boys each day was as follows: $$ \begin{array}{lrrrrrrrrrr} \hline \text { Ethan } & 9 & 24 & 8 & 9 & 5 & 8 & 9 & 10 & 8 & 10 \\ \hline \text { Drew } & 15 & 2 & 3 & 18 & 20 & 1 & 17 & 2 & 19 & 3 \\ \hline \end{array} $$ (a) Find the population mean and the range for the number of smallmouth bass caught per day by each fisherman. Do these values indicate any differences between the two fishermen's catches per day? Explain. (b) Draw a dot plot for Ethan. Draw a dot plot for Drew. Which fisherman seems more consistent? (c) Find the population standard deviation for the number of smallmouth bass caught per day by each fisherman. Do these values present a different story about the two fishermen's catches per day? Which fisherman has the more consistent record? Explain. (d) Discuss limitations of the range as a measure of dispersion.

Use the five test scores of 65,70,71 \(75,\) and 95 to answer the following questions: (a) Find the sample mean. (b) Find the median. (c) Which measure of central tendency better describes the typical test score? (d) Suppose the professor decides to curve the exam by adding 4 points to each test score. Compute the sample mean based on the adjusted scores. (e) Compare the unadjusted test score mean with the curved test score mean. What effect did adding 4 to each score have on the mean?

The acidity or alkalinity of a solution is measured using \(\mathrm{pH}\). A pH less than 7 is acidic, while a pH greater than 7 is alkaline. The following data represent the \(\mathrm{pH}\) in samples of bottled water and tap water. $$ \begin{array}{lllllll} \hline \text { Tap } & 7.64 & 7.45 & 7.47 & 7.50 & 7.68 & 7.69 \\ & 7.45 & 7.10 & 7.56 & 7.47 & 7.52 & 7.47 \\ \hline \text { Bottled } & 5.15 & 5.09 & 5.26 & 5.20 & 5.02 & 5.23 \\ & 5.28 & 5.26 & 5.13 & 5.26 & 5.21 & 5.24 \\ \hline \end{array} $$ (a) Which type of water has more dispersion in pH using the range as the measure of dispersion? (b) Which type of water has more dispersion in pH using the standard deviation as the measure of dispersion?

A certain type of concrete mix is designed to withstand 3000 pounds per square inch (psi) of pressure. The concrete is poured into casting cylinders and allowed to set for 28 days. The concrete's strength is then measured. The following data represent the strength of nine randomly selected casts (in psi). Compute the mean, median, and mode strength of the concrete (in psi). $$ 3960,4090,3200,3100,2940,3830,4090,4040,3780 $$

True or False: When comparing two populations, the larger the standard deviation, the more dispersion the distribution has. provided that the variable of interest from the two populations has the same unit of measure.
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