Final answer:
To address the questions posed by the student, one must calculate the Z-scores for the given heights, refer to the standard normal distribution to determine corresponding probabilities, and use these to calculate the probability of a variable falling within a certain range, as well as identifying the quartiles and specific height percentiles.
Step-by-step explanation:
The student's question is about calculating probabilities and quartiles for normally distributed variables. For a normally distributed variable X with a known mean and standard deviation, one can calculate the probability of X falling within a specified range using the standard normal distribution (Z-score).
Probability Calculation
To find the probability of a randomly chosen Asian adult male being between 65 and 69 inches tall:
- Calculate the Z-scores for 65 and 69 inches.
- Use the standard normal distribution table to find the probability corresponding to these Z-scores.
- The desired probability is the difference between these two probabilities.
Understanding Normal Distribution
In terms of meeting Asian adult males taller than 72 inches, we would calculate the Z-score for 72 inches and use the standard normal distribution table to determine the probability of a male being taller than 72 inches. Since the mean height is 66 inches with a standard deviation of 2.5 inches, a height of 72 inches is quite far from the mean, leading us to expect that very few males will be taller than this height.
Quartiles and Percentiles
The quartiles of a normal distribution divide the distribution into four equal parts. You would use the Z-scores corresponding to the 25th percentile (Q1) and the 75th percentile (Q3) to find these values. The middle 40 percent of heights would involve finding the Z-scores for the 30th and 70th percentiles and then converting these back to actual heights.