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Answer :
Final answer:
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.
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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
