Final answer:
To calculate how far below the mean a certain height is and how many standard deviations this is, we use the z-score. 71 inches is 8 inches below the mean and 2 standard deviations below. 75 inches is 4 inches below the mean and 1 standard deviation below.
Step-by-step explanation:
The heights of professional basketball players are normally distributed with a mean of 79 inches and a standard deviation of 4 inches. To find how far below the mean a certain height is and how many standard deviations this represents, we can use the concept of a z-score, which measures the number of standard deviations an observation is from the mean. For the first question:
- 71 inches is 8 inches below the mean (79 - 71 = 8).
- To calculate the number of standard deviations, we use the formula for the z-score: z = (X - μ) / σ. For 71 inches, z = (71 - 79) / 4 = -8 / 4 = -2. So, 71 inches is 2 standard deviations below the mean.
For the second question:
- 75 inches is 4 inches below the mean (79 - 75 = 4).
- Using the z-score formula, for 75 inches: z = (75 - 79) / 4 = -4 / 4 = -1. So, 75 inches is 1 standard deviation below the mean.