Answer :
Final answer:
a. The proportion of trucks expected to travel between 34,000 and 50,000 miles is approximately 40.82%. b. The percentage of trucks expected to travel less than 30,000 or more than 60,000 miles is approximately 26.6%. c. At least 80% of the trucks will travel at least 60,080 miles.
Explanation:
a. To find the proportion of trucks expected to travel between 34,000 and 50,000 miles, we need to calculate the z-scores for these values using the formula z = (x - μ) / σ where x is the value, μ is the mean, and σ is the standard deviation. The z-score for 34,000 miles is z = (34,000 - 50,000) / 12,000 = -1.33 and the z-score for 50,000 miles is z = (50,000 - 50,000) / 12,000 = 0. To find the proportion between these two values, we can use a standard normal distribution table. The area under the curve between -1.33 and 0 is approximately 0.4082 or 40.82%.
b. To find the percentage of trucks expected to travel less than 30,000 or more than 60,000 miles, we need to calculate the z-scores for these values. The z-score for 30,000 miles is z = (30,000 - 50,000) / 12,000 = -1.67 and the z-score for 60,000 miles is z = (60,000 - 50,000) / 12,000 = 0.83. To find the proportion outside these two values, we can subtract the area under the curve between -1.67 and 0.83 from 1. The area under the curve between -1.67 and 0.83 is approximately 0.734, so the percentage outside these two values is 1 - 0.734 = 0.266 or 26.6%.
c. To find the number of miles traveled by at least 80% of the trucks, we need to find the z-score that corresponds to the 80th percentile. We can find this value using a standard normal distribution table or a z-table. The z-score corresponding to the 80th percentile is approximately 0.84. We can then use the formula z = (x - μ) / σ to find the value of x. Substituting 0.84 for z, 0.84 = (x - 50,000) / 12,000. Solving for x, we get x = 0.84 * 12,000 + 50,000 = 60,080 miles.
d. If the standard deviation were 10,000 miles instead of 12,000 miles, the calculations for parts a, b, and c would be the same, except we would use the new standard deviation value in the calculations.
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