Final answer:
The probability that a package will be delivered in 18 hours or less is 84.13%. The guaranteed delivery time to be 90% sure is 18.84 hours.
Step-by-step explanation:
To find the probability that a package will be delivered in 18 hours or less, we need to find the area under the normal distribution curve to the left of 18 hours. This represents the percentage of packages that will be delivered within 18 hours.
Using Z-scores, we can calculate this probability as:
P(X ≤ 18) = P(Z ≤ (18 - 15)/3) = P(Z ≤ 1) = 0.8413
Therefore, the probability that a package will be delivered in 18 hours or less is 0.8413, or 84.13%.
To determine the guaranteed delivery time on all packages in order to be 90% sure that the package will be delivered before this time, we need to find the corresponding Z-score for the 90th percentile.
Using the Z-table or a calculator, we find that the Z-score for the 90th percentile is approximately 1.28.
Using the formula X = Zσ + μ, where X is the delivery time, Z is the Z-score, σ is the standard deviation, and μ is the mean, we can solve for the guaranteed delivery time:
X = (1.28 * 3) + 15 = 18.84
Therefore, the guaranteed delivery time on all packages in order to be 90% sure is 18.84 hours.