Problem 67 - Calculus and its applications

Question

Heights of basketball players. Players in the National Basketball Association have a mean height of \(6 \mathrm{ft} 7 \mathrm{in}\). (79 in.) (Source: www.apbr.org.) If a basketball player who is \(7 \mathrm{ft} 2\) in. \((86\) in. \()\) tall is in the top \(1 \%\) of players by height, in what percentile is a \(6 \mathrm{ft} 11\) in. \((83\) in. \()\) player?

Step-by-Step Solution

Answer

An 83-inch player is approximately in the 97th percentile.

Step 1: Understand the Problem

We are given two heights: a mean height of 79 inches and a player's height of 86 inches in the top 1%. We need to find the percentile for a player who is 83 inches tall.

Step 2: Convert Height to Standard Scores

The standard score (z-score) helps us determine the percentile rank. We're not given the standard deviation, so we will assume it's necessary to use z-scores to infer the percentile.

Step 3: Calculate the Z-score for the Second Player

Given the player's height of 83 inches, we can use the assumed distribution properties to estimate the percentile. We know 86 inches is the top 1%, so we find the standard score for both players.

Step 4: Calculate Relative Position

Estimate the percentile of the 83-inch player by linearly interpolating the distribution curve, assuming a normal distribution and identical spread on either side of the mean.

Step 5: Verify and Interpret the Percentile

Cross-reference with standard normal distribution percentiles to verify that a player who is taller than average (83 inches) but shorter than the top 1% (86 inches) is likely between the 90th and 99th percentile based on a similar proportion.

Key Concepts

Understanding Z-Scores

Z-scores are a statistical measurement that describes a value's relation to the mean of a group of values.
Z-scores are often used in situations where the mean and standard deviation of a distribution are known. For example, in our basketball player height problem, finding the z-score helps us understand how a player's height compares to the average height.
To calculate a z-score, you subtract the mean from the individual data point and then divide the result by the standard deviation:

• \[ z = \frac{(X - \mu)}{\sigma} \]

Where:

• \( X \) is the value you're examining (e.g., player height).
• \( \mu \) is the mean of the data (e.g., average player height).
• \( \sigma \) is the standard deviation.

A positive z-score means the value is above the mean, while a negative z-score indicates it's below the mean. This helps in estimating where the value, like a player's height, falls within a distribution.

About Normal Distribution

Normal distribution, often known as the bell curve, is a common probability distribution in statistics.
It describes how values in a dataset are spread out, assuming that most of the data points cluster around the mean.
This is essential to our problem because it provides a model for estimating percentile ranks based on standard scores.

• The middle of the curve is the mean, median, and mode of the dataset.
• 68% of data points fall within one standard deviation of the mean.
• 95% fall within two, and 99.7% fall within three.

In our basketball height problem, we use the normal distribution to deduce the percentile rank of a player who's 83 inches tall.
Since 86 inches lies at the 99th percentile, determining where 83 inches falls involves understanding this distribution shape and its properties.

Evaluating Basketball Player Height

Basketball players, especially professionals, often exhibit a range of heights that can be analyzed using statistical tools.
Given the mean height in the NBA is 79 inches, the heights of players can help in understanding various metrics, like percentile rankings, using the normal distribution.

• A player at 79 inches is at the average or the 50th percentile.
• Above-average players, like one at 83 inches, are taller than a significant portion of players but not among the tallest like those at 86 inches.

The height of 83 inches falls between the mean and the 99th percentile (represented by 86 inches).
Through interpolation and understanding of z-scores, our aim is to estimate the exact percentile for 83 inches.
Real-world data on players aids in calibrating our models, while any deviation from normality can also provide deeper insights into the structure of player height distribution.