Final answer:
a. The distribution of X is a normal distribution. b. The probability that a randomly selected car is traveling more than 81 mph is approximately 0.4502. c. The probability that a randomly selected car is traveling between 83 and 88 mph is approximately 0.4207. d. 75% of all cars travel at least 85.396 mph on the freeway.
Step-by-step explanation:
a. The distribution of X, which represents the speed of cars on the freeway, is a normal distribution.
b. To find the probability that a randomly selected car is traveling more than 81 mph, we need to calculate the z-score and then use a standard normal distribution table. The z-score can be calculated as (81 - 80) / 8 = 0.125. Using the standard normal distribution table, we find that the probability is approximately 0.4502.
c. To find the probability that a randomly selected car is traveling between 83 and 88 mph, we first calculate the z-scores for 83 and 88 as (83 - 80) / 8 = 0.375 and (88 - 80) / 8 = 1. The probability can be calculated by subtracting the area to the left of the z-score for 83 (0.375) from the area to the left of the z-score for 88 (1). Using the standard normal distribution table, we find that the probability is approximately 0.4207.
d. To find the speed at which 75% of all cars travel on the freeway, we need to find the z-score corresponding to the 75th percentile. Using the standard normal distribution table, we find that the z-score is approximately 0.6745. We can then use the z-score formula to find the corresponding speed: 0.6745 = (x - 80) / 8. Solving for x, we find that 75% of all cars travel at least 85.396 mph on the freeway.