HW 4 -- THE NORMAL LOADER
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-3.0
-2.0
-1.0
0.0
1.0
2.0
Z
X
3.0
First
Name
Last
Name
Total No.
of Letters
A cement manufacturing company has a machine that drops cement powder mix into
2,000-pound-capacity containers that are positioned beneath it on rail cars. The machine
has a dial marked "μ" to control the average (mean) amount of mix dispensed by the
machine into each container. The current setting is 2,000 pounds. Like most machines,
however, this loader has some variation in its performance. The amounts dispensed per
container are normally distributed with a standard deviation of 25 pounds plus the number
of letters in your official BU first and last names combined. The instructor’s loader, for
example, would have a standard deviation of 25+13=38 pounds. (Many real machines
actually have normally-distributed performance patterns.)
The amount that enters each container is determined by weighing it after filling. The
amount is stamped on the side of the container and the price of the filled container is set
accordingly. Thus, underfilled containers are not a problem. But if the loader dispenses
more than 2,000 pounds, the excess spills over the sides and is not recoverable.
1. Under current conditions, what percent (four significant figures) of the containers are
overfilled, resulting in loss of product?
___________%
2. If management wants the percentage of overfilled containers to be only 2.20000%
(0.0220000), to what setting (SIX significant figures) should the dial be moved?
____________lbs
3. Suppose the dial is stuck at 2,000 and cannot be moved. What size container (SIX
significant figures) should be used to achieve the 2.2% objective mentioned above?
____________lbs
4. Suppose the container supplier's largest container is 2,030 pounds. If these containers
are used, what percentage (four significant figures) of them will be overfilled?
____________%
5. Now the company is using the 2,030-pound containers, but the 2.2% objective is still
not being met. The machine maintenance person suggests "tuning up" the machine to
make it more consistent (reducing its standard deviation). To what value (four
significant figures) must the standard deviation be reduced to achieve the 2.2%
objective with the 2,030-pound containers?
____________lbs
On the next page, in each numbered section, enter the three known items of information
on the appropriate blanks, circle the unknown, then solve for the unknown (show
computation in box). In the normal-distribution diagram for each part, add SIX labels:
x, μ, z, σ, area, area shading. Turn in both pages. Due at the time of Quiz 4.
z = (X - μ) / σ
X=μ+zσ
μ=X-zσ
σ = (X - μ) / z
1. z = __________
0 .4 5
0 .4
0 .3 5
x = __________
0 .3
0 .2 5
0 .2
μ = __________
0 .1 5
0 .1
0 .0 5
σ = __________
0
- 3 .0
- 2 .0
- 1 .0
0 .0
1 .0
2 .0
Z
3 .0
X
0 .4 5
2. z = __________
0 .4
0 .3 5
x = __________
0 .3
0 .2 5
0 .2
μ = __________
0 .1 5
0 .1
σ = __________
0 .0 5
0
- 3 .0
- 2 .0
- 1 .0
0 .0
1 .0
2 .0
Z
3 .0
X
0 .4 5
3. z = __________
0 .4
0 .3 5
x = __________
0 .3
0 .2 5
0 .2
μ = __________
0 .1 5
0 .1
σ = __________
0 .0 5
0
- 3 .0
- 2 .0
- 1 .0
0 .0
1 .0
2 .0
Z
3 .0
X
0 .4 5
4. z = __________
0 .4
0 .3 5
x = __________
0 .3
0 .2 5
0 .2
μ = __________
0 .1 5
0 .1
σ = __________
0 .0 5
0
- 3 .0
- 2 .0
- 1 .0
0 .0
1 .0
2 .0
Z
3 .0
X
0 .4 5
5. z = __________
0 .4
0 .3 5
0 .3
x = __________
0 .2 5
0 .2
μ = __________
0 .1 5
0 .1
0 .0 5
σ = __________
0
- 3 .0
- 2 .0
- 1 .0
0 .0
1 .0
2 .0
Z
X
3 .0