1. Using the Standard Normal Distribution to Solve SPC Problems
Table A.1: “Areas Under the Normal Curve”
ISE 327 Ch. 3 Homework

2. Applications of the Normal Distribution
A certain machine makes electrical resistors having a mean resistance of 40 ohms and a standard deviation of 2 ohms. What percentage of the resistors will have a resistance less than 44 ohms?
Solution: X is normally distributed with μ = 40 and σ = 2 and x = 44
P(X<44) = P(Z< +2.0) =
Therefore, we conclude that 97.72% will have a resistance less than 44 ohms What percentage will have a resistance greater than 44 ohms?
ISE 327 Ch. 3 Homework

3. The Normal Distribution “In Reverse”
Given a normal distribution with μ = 40 and σ = 6, find the value of X for which 45% of the area under the normal curve is to the left of X.
If P(Z < k) = 0.45, then
k = (table)
Z = / (x-µ) / σ
= (X – 40)/6
4) X- 40 = 6 * (-0.125)
Therefore, X = 39.25
\ \ \ 45%
k =
X = 39.25
ISE 327 Ch. 3 Homework

4. Problem 3.9a Part 1 Specifications 0.500 ± 0.025
LSL = USL = 0.525
Normal distribution
μx = and σx =
Find the percent that are within
specifications.
Step 1 Find the corresponding z values.
Z = (x-µ) / σ
ZL = ( )/0.0065
ZL = -4.62
Z = (x-µ) / σ
ZU = ( )/0.0065
ZU = 3.08
LSL = 0.475
USL = 0.525
ZL = -4.62
ZU = 3.08
ISE 327 Ch. 3 Homework

5. Problem 3.9a Part 2 Specifications 0.500 ± 0.025
LSL = USL = 0.525
Normal distribution
μx = and σx =
Find the percent that are within specifications.
Step 2 Find the area under the curve between the LSL and USL.
P(Z  3.08) - P(Z  -4.62)
= – 0.0 =
Therefore % are within specifications.
What percent are outside of specifications?
LSL = 0.475
USL = 0.525
99.897%
ZL = -4.62
ZU = 3.08
ISE 327 Ch. 3 Homework

6. Problem 3.9b Specifications 0.500 ± 0.025 LSL = 0.475 USL = 0.525
Normal distribution
Process variability reduced to 0.003
μx = and σx = 0.003
Find the percent that are within specifications.
Z = (x-µ) / σ
ZL = ( )/0.003
ZL = -10.0
Similarly, ZU = 6.667
P(Z  6.667) - P(Z  -10.0)
Approximately 100%
Therefore 100% are within specifications.
LSL = 0.475
USL = 0.525
~ 100%
ZL = -10.0
ZU = 6.667
ISE 327 Ch. 3 Homework