Final answer:
The probability of a mail carrier taking at least 350 minutes for a route is found by calculating the Z-score and using it to determine the area under the normal curve to the right of the Z-score value.
Step-by-step explanation:
The student's question involves finding the probability that a mail carrier will cover his route in at least 350 minutes, given that the average (mean) time is 380 minutes and the standard deviation is 16 minutes, with the assumption that the variable is normally distributed. To solve this, one would use the Z-score formula, which is Z = (X - μ) / σ, where X is the value of interest (350 minutes), μ is the mean (380 minutes), and σ is the standard deviation (16 minutes).
To calculate the Z-score:
- Subtract the mean from the value of interest: 350 - 380 = -30.
- Divide the result by the standard deviation: -30 / 16 = -1.875.
- Use a Z-table or statistical software to find the probability corresponding to Z = -1.875.
- Since we want the probability of at least 350 minutes, we look for P(Z > -1.875).
The probability obtained from the Z-table would be the area under the normal curve to the right of Z = -1.875. This value would give us the probability that the mail carrier takes at least 350 minutes to cover his route.