Final answer:
To find the proportion of trucks traveling between two distances and the percentage of trucks traveling less than one distance or more than another, we use the z-score formula and a standard normal distribution table or calculator.
Step-by-step explanation:
a. To find the proportion of trucks expected to travel between 35 and 50 thousand miles in a year, we need to calculate the z-scores for both distances using the formula z = (x - mean) / standard deviation. So for 35 thousand miles, the z-score is (35 - 50) / 11 = -1.36, and for 50 thousand miles, the z-score is 0. Using a standard normal distribution table or a calculator, we can find that the proportion of trucks between these two distances is the difference between the cumulative probabilities associated with the z-scores. So the proportion is P(Z < 0) - P(Z < -1.36) = 0.5 - 0.0853 = 0.4147, or approximately 41.47%.
b. To find the percentage of trucks expected to travel either less than 40 thousand or more than 65 thousand miles in a year, we need to calculate the z-scores for both distances. For 40 thousand miles, the z-score is (40 - 50) / 11 = -0.9091, and for 65 thousand miles, the z-score is (65 - 50) / 11 = 1.3636. Again, using a standard normal distribution table or a calculator, we can find the cumulative probabilities associated with these z-scores. The percentage of trucks expected to travel less than 40 thousand or more than 65 thousand miles is the sum of these probabilities, which is P(Z < -0.9091) + P(Z > 1.3636). The total percentage is approximately 35.13%.