Assume that the mean hourly cost to operate a commercial airplane follows the normal distribution with a mean of $2,325 per hour and a standard deviation of $240.
What is the operating cost for the lowest 5 percent of the airplanes?
Z = (score-mean)/SD
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (.05) and its Z score. Put values into above equation and solve for raw score.
1549
To find the operating cost for the lowest 5 percent of the airplanes, we need to calculate the z-score corresponding to the 5th percentile (or 0.05 percentile) of a normal distribution.
Step 1: Calculate the z-score
The z-score formula is given by: Z = (X - μ) / σ
Where:
Z is the z-score,
X is the value we are interested in (the operating cost),
μ is the mean of the distribution, and
σ is the standard deviation of the distribution.
In this case, we want to find the z-score for the 5th percentile, which means our desired area under the curve is 0.05. Since the normal distribution is symmetrical, we can find the z-score by looking up the z-value that corresponds to an area of 0.05 to the left of it in the standard normal distribution table.
Step 2: Look up the z-score
By looking up the z-score in the standard normal distribution table (also known as the z-table), we find that the z-score corresponding to a 5th percentile is approximately -1.645.
Step 3: Calculate the operating cost
Now that we have the z-score, we can use it to calculate the operating cost for the lowest 5 percent of the airplanes.
Operating cost = μ + (Z * σ)
= $2,325 + (-1.645 * $240)
= $2,325 - $395.80
= $1,929.20
Therefore, the operating cost for the lowest 5 percent of the airplanes is approximately $1,929.20.