Unit 4
The Normal Curve and
Normal Approximation
FPP Chapter 5
x-
f X (x) = 1 exp [- 1 ( ) ] for - < x < 
2 
 2
2
0.4
0.3
0.2
0.1
0.0 -4 -3 -2 -1
0
X
1
2
3
4
1. Symmetric about zero
2. Area under the curve = 100%
3. Always above the horizontal axis
4. Area between -1 and +1 is about 68%.
Area between -2 and +2 is about 95%.
4-1
Normal
A.05
-30 -20 -10 0 10 20
30
X
0
10
20
30
40
50
60
70
80
X
Standard units say how many SD’s above or
below the average a value is.
They allow us to compare different normal
curves.
4-2
Normal
Normal Approximation
For many lists, the % entries falling into an interval can be
estimated using the normal curve.
(1) Convert the interval in question to standard units.
(2) Find the area above this interval, under the normal
curve.
That area is approximately the % entries from the list falling
into the interval.
0.4
0.3
0.2
0.1
0.0
? ? ? ? ? ? ? ? ?
-4
-3 -2
-1
0
1
Original Units
2
3
4
Standard Units
4-3
Normal
Example: Exam Scores
Average score = 20 points
SD = 5 points
You score 25 points.
Is your score above or below average?
How many points above or below average?
How many SD’s is that?
25 points is ___________ in standard units.
Convert the score 17 21
points to standard units.
What is the score 20 points in standard units?
4-4
Normal
Reading the Normal Table
4-5
Normal
4-6
Normal
Normal Curve Arithmetic
4-7
Normal
Normal Curve Arithmetic
4-8
Normal
Service Times
The time to complete the 40,000 mile service at a
local automobile dealership follows a normal
curve with average 100 minutes and SD 10
minutes.
What is the probability that it will take between
100 and 115 minutes?
You bring your car in for service, but you need it
to be done in 120 minutes or less. What is the
chance that your service will be done in 120
minutes or less?
4-9
Normal
Percentiles
Refer back to the Exam Scores example:
What is the 90th percentile?
What is the 10th percentile?
4-10
Normal
Some Normal Curve
Problems
1. The diameters of metal rods manufactured by
a certain supplier follow a normal distribution
with mean 4.0 centimeters and SD 0.2
centimeters.
(a) What proportion of the rods have diameters
less than 3.8 cm?
(b) What proportion of the rods have diameters
greater than 4.2 cm?
(c) What proportion of the rods have diameters
between 3.9 and 4.1 cm?
2. A consultant states that her uncertainty about
the time needed to complete a construction
project can be represented by a normal random
variable with mean 60 weeks and SD 8 weeks.
(a) What is the probability that the project will
take more than 70 weeks to complete?
(b) What is the probability that the project will
take less than 52 weeks to complete?
4-11
(c) What is the probability that the
Normal
project will take between 52 and 70
weeks to complete?
More Normal Curve
Problems
3. A company services gas central-heating
furnaces. A review of its records indicates that
the time taken for a routine maintenance
service call can be represented by a normal
distribution with mean 60 minutes and SD 10
minutes.
(a) What proportion of such service calls take
more than 45 minutes?
(b) What proportion of such service calls take
less than 75 minutes?
(c) Sketch a graph to illustrate the reason for
the coincidence in the answers to (a) and (b).
4. On average, graduates of a particular
university earn $59,000 five years after
graduation. The standard deviation is $4,000.
What percent earn less than $54,000? What
percent earn more than $70,000? What
assumptions did you make? Are those
assumptions reasonable?
4-12
Normal
Even More Practice Problems
5. Trucks at a certain warehouse are loaded with
200 boxes each. It is known that, on average,
8% of all boxes are damaged during loading.
What percent of the time do 21 or more boxes
get damaged if the SD is 3.8 boxes? What
assumptions did you make?
6. I am considering two alternative investments.
In both cases, I am unsure about the
percentage return but believe that my
uncertainty can be represented by normal
distributions with means and SD's as follows.
For Investment A, the mean is 10.4, the SD is
1.2. For Investment B, the mean is 11.0, the
SD is 4.0. I want to make the investment that
is more likely to produce a return of at least
10%.
Which should I choose?
4-13
Normal
More Practice Problems
7. The mean GPA of University of Washington's
last graduating class was 2.7 with SD 0.4.
What GPA did the 90th percentile have?
8. In a 3 year period, 665,281 people took the
GMAT (including repeaters). The distribution
of scores approximately follows a normal
curve. The mean score was 492 and the SD
was 103. What was the 80th percentile for
these GMAT scores? What proportion of
scores were above 550?
4-14
Normal
More Normal Curve Problems
9. It is estimated that major league baseball
game times to completion follow a normal
distribution with mean 132 minutes and SD 12
minutes.
(a) What proportion of all games last between
120 and 150 minutes?
(b) Thirty-three percent of all games last longer
than how many minutes?
(c) What is the 67th percentile of game times to
completion?
(d) What proportion of games last less than 120
minutes?
10. The weights of the contents of boxes of a
brand of cereal have a normal distribution with
mean 24 ounces and SD 0.7 ounces.
(a) What is the probability that the contents of a
randomly chosen box weigh less than 23
ounces?
(b) The contents of 10% of all boxes weight
more than how many ounces?
4-15
Normal
(c) What proportion of boxes have
contents weighing between 23.5
and 24.5 ounces?
11. A management consultant found that the
amount of time per day spent by executives
performing tasks that could be done equally
well by subordinates followed a normal
distribution with mean 2.4 hours. It was also
found that 10% of executives spent over 3.5
hours per day on such tasks. Find the
standard deviation of the distribution of daily
time spent by executives on tasks of this type.
(Newb228)
12. The cereal manufacturer of problem 10
wants to adjust the production process so that
the mean weight of the contents of the boxes
of cereal is still 24 ounces, but only 3% of the
boxes will contain less than 23 ounces of
cereal. What SD for the weights of the
contents is needed to attain this objective?
4-16
Normal
13. A video display tube for computer graphics
terminals has a fine mesh screen behind the
viewing surface. During assembly the mesh is
stretched and welded onto a metal frame. Too
little tension at this stage will cause wrinkles,
while too much tension will tear the mesh. The
tension is measured by an electrical device
with output readings in millivolts (mV). At the
present time, the tension readings for
successive tubes follows a normal distribution
with mean 275 and standard deviation 43 mV.
(a) The minimum acceptable tension
corresponds to a reading of 200 mV. What
proportion of the tubes exceed this limit?
(b) The mean tension can be adjusted in the
production process, but the SD remains at 43
mV regardless of the mean tension setting.
What mean tension setting should be used to
make it so that 2% of the tubes have tension
readings below the limit of 200 mV?
(c) In production, tension above 375 mV will
usually tear the mesh. Thus the acceptable
4-17
range of tension readings is actually
Normal
200 mV to 375 mV. What proportion
of tubes are in this acceptable range?