Major League Baseball now records information about every pitch thrown in every game of every season. Statistician Jim Albert compiled data about every pitch thrown by 20 starting pitchers during the 2009 MLB season. The data set included the type of pitch thrown (curveball, changeup, slider, etc.) as well as the speed of the ball as it left the pitcher’s hand. A histogram of speeds for all 30,740 four-seam fastballs thrown by these pitchers during the 2009 season is shown below, from which we can see that the speeds of these fastballs follow a Normal model with mean μ = 92.12 mph and a standard deviation of σ = 2.43 mph. Compute the z-score of pitch with speed 92.7 mph. (Round your answer to two decimal places.)

The **z-score** is a measure of how many standard deviations a particular value is from the mean in a normally distributed data set. It is calculated using the formula:

\[
z = \frac{x - \mu}{\sigma}
\]

Where:
- \(x\) is the value for which we're calculating the z-score (in this case, 92.7 mph),
- \(\mu\) is the mean (92.12 mph),
- \(\sigma\) is the standard deviation (2.43 mph).

Substitute the values into the formula:

\[
z = \frac{92.7 - 92.12}{2.43}
\]

\[
z = \frac{0.58}{2.43} \approx 0.24
\]

So, the z-score for a pitch with a speed of 92.7 mph is approximately **0.24**.

Would you like further details on how z-scores work or any related concepts? Here are five follow-up questions to consider:

1. What does a z-score of 0.24 signify in the context of the distribution of fastball speeds?
2. How would you interpret the z-score of a pitch significantly faster, say at 97 mph?
3. What percentage of pitches are expected to have speeds less than 92.7 mph based on the z-score?
4. How do different types of pitches (e.g., curveballs) compare in speed distribution to fastballs?
5. Why is it useful for statisticians to model pitch speeds using a normal distribution?

**Tip**: A z-score of 0 means the value is exactly at the mean of the distribution!

Calculate the z-score for a fastball with a speed of 92.7 mph from the 2009 MLB season, using the normal distribution model with a mean of 92.12 mph and a standard deviation of 2.43 mph. Suitable for students in Grades 9-12, this problem explains the concept of z-scores and their role in understanding pitch speed variation.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation