Questions: Trucks in a delivery fleet travel a mean of 100 miles per day with a standard deviation of 38 miles per day. The mileage per day is distributed normally. Find the probability that a truck drives between 119 and 149 miles in a day. Round your answer to four decimal places.

Trucks in a delivery fleet travel a mean of 100 miles per day with a standard deviation of 38 miles per day. The mileage per day is distributed normally. Find the probability that a truck drives between 119 and 149 miles in a day. Round your answer to four decimal places.

Transcript text: Trucks in a delivery fleet travel a mean of 100 miles per day with a standard deviation of 38 miles per day. The mileage per day is distributed normally. Find the probability that a truck drives between 119 and 149 miles in a day. Round your answer to four decimal places.

Solution

Solution Steps

Step 1: Define the Normal Distribution Parameters

The trucks in the delivery fleet have a normal distribution of daily mileage with the following parameters:

Mean (\( \mu \)): 100 miles
Standard Deviation (\( \sigma \)): 38 miles

Step 2: Calculate Z-scores

To find the probability that a truck drives between 119 and 149 miles, we first calculate the Z-scores for the lower and upper bounds.

For the lower bound (119 miles): \[ Z_{start} = \frac{119 - 100}{38} = 0.5 \]

For the upper bound (149 miles): \[ Z_{end} = \frac{149 - 100}{38} \approx 1.2895 \]

Step 3: Calculate the Probability

Using the Z-scores, we can find the probability that a truck drives between 119 and 149 miles in a day. This is given by: \[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(1.2895) - \Phi(0.5) \] From the calculations, we find: \[ P \approx 0.2099 \]