Final answer:
When designing the placement of a CD player in a new model car, engineers must consider the forward grip reach of the driver. Using the Empirical Rule, we can calculate the percentage of scores greater than a given value, less than a given value, and between two given values. In this case, we found that approximately 84.13% of scores are greater than 24.1 inches, 97.72% are less than 28.9 inches, and 81.85% are between 22.5 and 27.3 inches.
Step-by-step explanation:
When designing the placement of a CD player in a new model car, engineers must consider the forward grip reach of the driver. In this case, we are given that the forward grip reaches of women are normally distributed with a mean of 25.7 inches and a standard deviation of 1.6 inches. We can use the Empirical Rule (68-95-99.7 rule) to find the indicated quantities.
a) To find the percentage of scores greater than 24.1 inches, we need to calculate the z-score for 24.1 using the formula z = (x - μ) / σ, where x is the given score, μ is the mean, and σ is the standard deviation. The z-score is (24.1 - 25.7) / 1.6 = -1. We can then use a standard normal distribution table or a z-score calculator to find the percentage of scores greater than -1, which is approximately 84.13%.
b) To find the percentage of scores less than 28.9 inches, we calculate the z-score for 28.9 using the same formula. The z-score is (28.9 - 25.7) / 1.6 = 2. The percentage of scores less than 2 is approximately 97.72%.
c) To find the percentage of scores between 22.5 and 27.3 inches, we need to calculate the z-scores for both values and find the area between them. The z-score for 22.5 is (22.5 - 25.7) / 1.6 = -2 and the z-score for 27.3 is (27.3 - 25.7) / 1.6 = 1. The area between these z-scores can be found using a standard normal distribution table or a calculator, which is approximately 81.85%.