The time that it takes a randomly selected job applicant to perform a certain task has a distribution that can be approximated by a normal distribution with a mean value of 150 sec and a standard deviation of 20 sec. The fastest 10% are to be given advanced training. What task times qualify individuals for such training? (Round your answer to one decimal place.)
To find the task times that qualify individuals for advanced training, we need to find the value that corresponds to the 10th percentile of the normal distribution.
Step 1: Calculate the z-score corresponding to the 10th percentile.
The z-score formula is given by:
z = (x - μ) / σ
where z is the z-score, x is the task time, μ is the mean, and σ is the standard deviation.
To find the z-score corresponding to the 10th percentile, we need to look up the z-score in the standard normal distribution table (also known as the z-table). The 10th percentile corresponds to a cumulative probability of 0.10, which is equivalent to a z-score of -1.28.
Step 2: Solve for x (task time).
Using the z-score formula, we can rearrange it to solve for x:
x = z * σ + μ
Plugging in the values:
x = -1.28 * 20 + 150 = 124
So, any task times less than or equal to 124 seconds would qualify individuals for advanced training.
Therefore, the individuals who complete the task in 124 seconds or less qualify for advanced training.
To find the task times that qualify individuals for advanced training, we need to find the cutoff point where only 10% of the applicants are faster.
Step 1: Calculate the z-score corresponding to the desired percentile.
The z-score formula is given by: z = (x - μ) / σ, where x is the task time, μ is the mean, and σ is the standard deviation.
In this case, we want to find the z-score corresponding to the 90th percentile (since we are looking for the fastest 10%). We can use a standard normal distribution table or a calculator to find the z-score corresponding to the 90th percentile, which is approximately 1.28.
Step 2: Rearrange the z-score formula to solve for x.
x = (z * σ) + μ
Substituting the known values, we have:
x = (1.28 * 20) + 150
x ≈ 175.6
So, the task times that qualify individuals for advanced training are approximately 175.6 seconds or less.
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 (.10) and its related Z score. Insert with other data in equation above and solve.