Express Courier Service has found that the delivery time for packages is normally distributed, with a mean of 13 hours and a standard deviation of 2 hours.

(a) For a package selected at random, what is the probability that it will be delivered in 18 hours or less? (Round your answer to four decimal places.)

(b) What should be the guaranteed delivery time on all packages in order to be 95% sure that the package will be delivered before this time? (Hint: Note that 5% of the packages will be delivered at a time beyond the guaranteed time period.) (Round your answer to one decimal place.)

The probability that a package will be delivered in 18 hours or less is approximately 0.9938. To be 95% sure that a package will be delivered before a given time, the guaranteed delivery time should be calculated using the Z-score corresponding to a cumulative probability of 0.95.

Explanation:

To find the probability that a package will be delivered in 18 hours or less, we need to calculate the Z-score for 18 hours using the given mean and standard deviation. The formula for calculating the Z-score is: Z = (X - mean) / standard deviation. Substituting the given values, we get: Z = (18 - 13) / 2 = 2.5.

The next step is to find the corresponding probability from the Z-table. Looking up the Z-score of 2.5 in the Z-table, we find that the probability is approximately 0.9938. Therefore, the probability that a randomly selected package will be delivered in 18 hours or less is 0.9938 (rounded to four decimal places).

To determine the guaranteed delivery time that ensures a 95% probability of delivery before that time, we need to find the Z-score that corresponds to a cumulative probability of 0.95. From the Z-table, the Z-score for a cumulative probability of 0.95 is approximately 1.645.

Using the formula Z = (X - mean) / standard deviation, we can solve for X (the guaranteed delivery time): 1.645 = (X - 13) / 2. Rearranging the equation, we get: X - 13 = 1.645 * 2. Solving for X, we find: X = 13 + 1.645 * 2 = 16.29. Therefore, the guaranteed delivery time should be 16.3 hours (rounded to one decimal place) in order to be 95% sure that the package will be delivered before that time.

Favouredlyf Favouredlyf · Answer 2 · 2025-04-30T03:54:42-04:00

Answer:

Step-by-step explanation:

Express Courier Service has found that the delivery time for packages is normally distributed. So we use

z = (x - mean)/standard deviation

mean = 13

standard deviation = 2

x = time of delivery in hours

a) P(0 lesser than/equal to 18)

z = (18-13)/2 =5/2 = 2.5

Using the normal distribution table, the value is 0.9938

b) to be 95% sure, let the time be t

From the table, the equivalent of z that is 0.95 = 1.645

So 1.645 = (t-13)/2

t-13 = 3.29

t = 3.29+13= 16.29 hours