The inner diameter of a piston is normally distributed with mean of 10 cm and standard deviation 0.03cm
a)below what value of inner diameter will fall fifteen percent of the piston rings?
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 (.15) related to a Z score. Insert the values into the above equation.
To find the value below which fifteen percent of the piston rings will fall, we need to find the corresponding z-score and then convert it back to the original unit of measurement (cm in this case).
Step 1: Find the z-score
The z-score represents the number of standard deviations a given value is from the mean in a normal distribution. We can use the standard normal distribution table or a statistical calculator to find the z-score corresponding to the given percentage.
In this case, since we want to find the value below which fifteen percent of the distribution falls, we need to find the z-score representing the cumulative probability of 0.15.
Step 2: Convert z-score to actual value
Once we have the z-score, we can use it to find the corresponding value in the original measurement unit (cm).
The formula to convert a z-score to an actual value is:
x = μ + (z * σ)
Where:
x = actual value
μ = mean
z = z-score
σ = standard deviation
Let's calculate the value below which fifteen percent of the piston rings will fall:
Step 1: Find z-score:
Using a standard normal distribution table or a calculator, we find that the z-score corresponding to a cumulative probability of 0.15 is approximately -1.04.
Step 2: Convert z-score to actual value:
x = 10 + (-1.04 * 0.03)
x = 10 - 0.0312
x ≈ 9.9688 cm
Therefore, below a diameter of approximately 9.9688 cm, fifteen percent of the piston rings will fall.