Final answer:
Z-scores are calculated to determine how individual measurements compare to the average within a normal distribution. A height of 77 inches is 0.5141 standard deviations below the mean, while 85 inches is 1.5424 standard deviations above. A reported z-score of 3.5 for a player's height is very unusual and might not be readily believable.
Step-by-step explanation:
The question involves using z-scores to understand where individual measurements fall within a normal distribution relative to the mean and standard deviation. The z-score is calculated by subtracting the mean from the measurement and then dividing by the standard deviation.
Calculation of Z-scores
To calculate the z-score for a height of 77 inches (a), subtract the mean (79 inches) from 77 and then divide by the standard deviation (3.89 inches):
z = (77 - 79) / 3.89 = -0.5141
A z-score of -0.5141 means that 77 inches is 0.5141 standard deviations below the mean height.
For a height of 85 inches (b), calculate the z-score as follows:
z = (85 - 79) / 3.89 = 1.5424
This implies an NBA player with a height of 85 inches is taller than the average, being 1.5424 standard deviations above the mean.
If a player reports a z-score of 3.5, this would be highly unusual because it means the player's height is 3.5 standard deviations above the mean. Given that a z-score of 3 or more is considered rare in a normal distribution, it might warrant further scrutiny before being accepted as accurate (c).