Express Courier Service has found that the delivery time for packages is normally distributed with mean 14 hours and standard deviation 2 hours. 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 guarantee time period)
It would seem that you are only concerned with packages that would take longer than normal time to delivered. "Beyond" could infer either direction.
95% = mean + 1.645 SD
Insert value and solve.
If it were either direction:
95% = mean ± 1.96 SD
To determine the guaranteed delivery time, we need to find the value, denoted as X, such that only 5% of packages will be delivered beyond this time.
Since the delivery time is normally distributed with a mean of 14 hours and a standard deviation of 2 hours, we can use the standard normal distribution (Z-score) to find the value of X.
The Z-score is calculated using the formula: Z = (X - μ) / σ
Here, μ represents the mean (14 hours) and σ represents the standard deviation (2 hours).
To find the Z-value corresponding to the 5th percentile (the value at which 5% of packages will be delivered beyond), we can use a standard normal distribution table or calculator.
The Z-value for the 5th percentile is -1.645.
Now, we can solve for X in the formula:
-1.645 = (X - 14) / 2
Multiplying both sides by 2 gives:
-3.29 = X - 14
Adding 14 to both sides yields:
X = 14 - 3.29
Calculating this expression gives:
X ≈ 10.71
Therefore, the guaranteed delivery time on all packages should be approximately 10.71 hours in order to be 95% sure that the package will be delivered before this time.
To find the guaranteed delivery time that ensures a 95% probability of delivering packages before that time, we need to find the z-score associated with the 95th percentile of the standard normal distribution.
The z-score represents the number of standard deviations a value is away from the mean in a normal distribution. It can be calculated using the formula:
z = (x - mean) / standard deviation
In this case, the mean (μ) is 14 hours and the standard deviation (σ) is 2 hours. We want to find the z-score that corresponds to the 95th percentile, which corresponds to a cumulative probability of 0.95.
Using a table of z-scores or a calculator, we can find that the z-score for a cumulative probability of 0.95 is approximately 1.645.
Now we can use the z-score formula to find the guaranteed delivery time (x):
1.645 = (x - 14) / 2
Solving for x, we have:
1.645 * 2 = x - 14
x = 1.645 * 2 + 14
x ≈ 3.29 + 14
x ≈ 17.29
Therefore, the guaranteed delivery time on all packages should be approximately 17.29 hours to be 95% sure that the package will be delivered before this time.