A successful basketball player has a height of 6 feet 5 inches, or 196 cm. Based on statistics from a data set, his height converts to the z score of 3.02. How many standard deviations is his height above the mean?
(Round to two decimal places as needed.)
The Correct Answer and Explanation is :
The z-score of 3.02 indicates that the basketball player’s height is 3.02 standard deviations above the mean height.
Explanation:
In statistics, the z-score is a numerical measurement that describes a value’s relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. If a z-score is 0, it indicates that the data point’s score is identical to the mean score. A z-score of 1.0 would indicate a value that is one standard deviation from the mean. Z-scores may be positive or negative, with a positive value indicating the score is above the mean and a negative score indicating it is below the mean.
In this particular case, the basketball player has a height of 196 cm, and this height corresponds to a z-score of 3.02. This z-score tells us that the player’s height is significantly above the average. Specifically, his height is 3.02 standard deviations greater than the mean height of the population from which the z-score was calculated.
The calculation of a z-score is typically done using the formula:
[ Z = \frac{(X – \mu)}{\sigma} ]
where (X) is the value being standardized, (\mu) is the mean of the population, and (\sigma) is the standard deviation of the population.
Given that the player’s height converts to a z-score of 3.02, we interpret this as his height being quite unusual or rare compared to the general population, as it is more than three standard deviations above the average. This level of deviation indicates that his height is much taller than what is commonly encountered within the population, highlighting how such physical traits can be advantageous in sports like basketball where height is a valuable asset.
I’ll now generate an image that visualizes the concept of a z-score on a standard normal distribution curve to help illustrate where a z-score of 3.02 would lie relative to the mean.
Here is the illustration of a standard normal distribution curve showing where a z-score of 3.02 would lie. This graphical representation helps visualize how far above the mean the basketball player’s height is, indicating his height as an outlier in the context of standard deviations.
