A tractor manufacturing company machines axles to extremely small tolerances. The main power axles have a mean diameter of 3.0000 inches and a standard deviation of 0.0050 inch. What is the probability that a given axle will have a diameter between 2.9915 and 3.0004 inches? Assume that errors are normally distributed.
Using the computational tool at
http://psych.colorado.edu/~mcclella/java/normal/accurateNormal.html ,
I get 48.7%
I need help understanding how to solve by going through step by step. I do not want to know the answer, I need to learn the process. Please help me through that.
You need a table of the normal distribution or "error" function, e^-x^2. This involves an integral that must be calculated "by hand" because there is not simple closed-form integral other than the error function itself. The web site I gave you performs this calculation automatically.
For a step by step explanation, see
http://stattrek.com/Tables/Normal.aspx
To find the probability that a given axle will have a diameter between 2.9915 and 3.0004 inches, we need to use the concept of the standard normal distribution.
Step 1: Standardizing the values
We need to find the Z-scores for the given diameter values. The Z-score formula is:
Z = (X - μ) / σ
Where:
Z = Z-score
X = Diameter value
μ = Mean diameter
σ = Standard deviation
For the lower limit, X = 2.9915, so we calculate the Z-score:
Z1 = (2.9915 - 3.0000) / 0.0050
For the upper limit, X = 3.0004, so we calculate the Z-score:
Z2 = (3.0004 - 3.0000) / 0.0050
Step 2: Finding the probabilities
Once we have the Z-scores, we need to find the probabilities associated with those values. We can use a standard normal distribution table or a calculator/ software to find the probabilities.
From the table or software, we find the values corresponding to the Z-scores. Let's say Z1 corresponds to P1 and Z2 corresponds to P2.
Step 3: Calculating the probability
To find the probability between the two limits, we subtract the smaller probability from the larger probability:
Probability = P2 - P1
This will give us the desired probability.
Note: Make sure to use the correct table for the Z-scores if you are using a standard normal distribution table.
Once you have the Z-values, you can easily calculate the probability using a calculator or software like Excel or Python. Alternatively, you can use online tools that calculate normal distribution probabilities, such as probability calculators or statistical software.