completely determined by its mean
and
standard deviation
and, as a result, the standard deviation
is
a natural "yardstick"jor a normal distribution
In
any nonnal distribution, the same percentage
of
items will fall within some specified number
of
standard deviations
of
the mean. We can find
"what%
of
items fall where"
in
a normal
distribution simply by expressing our variable
X's
value
in
terms
of"how
many standard
deviations away from the mean it
is
Xf.l
7'~
i.e., the "standard normal variable"
Z
=
(J
~

f
C
2
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Exercise: Normal Distribution
Respond
to
the
following
questions:
1.
I'd speculate that the "height of adult males in Minnesota" could be modeled accurately by a normal distribution with a mean of 70 inches, and a standard deviation of 3 inches. Based on these assumptions, what % of males are taller than 76 inches? Only 10% of males are taller than ?
2.
Suppose
packages
of
cream
cheese
coming
from
an
automated
processor
have
weights
that
are
normally
distributed.
As
the
process
is
presently
operating,
the
mean
package
weight
is
8.2
ounces
and
the
standard
deviation
of
package
weights
is
.1
ounce.
a.
If
the
packages
of
cream
cheese
are
labeled
"8
ounces",
what
proportion
of
the
packages
weigh
less
than
the
labeled
amount?
b.
What
would
the
standard
deviation
need
to
be
reduced
to,
so
that
only
.5%
of
the
packages
weigh
less
than
the
labeled
amount?
c.
Suppose
they
are
unable
to
reduce
the
variability
in
this
process,
but
they
still
want
only
.5%
of
the
packages
to
weigh
less
than
the
labeled
amount.
To
what
level
do
they
need
to
increase
the
mean
amount
per
package
in
order
to
accomplish
this?
3
SAMPLING DISTRIBUTIONS
Consider
the
very
simplistic
population
made
up
of
only
the
5
numbers
[1,
2,
3,
4,
5]
Suppose
we
select
a
simple
random
sample
of
2
items
from
this
population:
Possible
Outcomes
1,
2
1,
3
1,
4
1,
5
2,
3
2,
4
2,
5
3,
4
31
5
4,
5
Possible
sample
Probability
1.0
means
Probability
1.5
2.0
2.5
3.0
3.5
4 . 0
4.5
Possible
sample
means
1.5
2.0
2.5
3.0
2.5
3.0
3.5
3.5
4.0
4.5
What
we've
developed
here
is
a
list
of
all
the
possible
values
for
a
variable,
and
the
probabilities
associated
with
each
of
those
possible
values;
i.e.
this
is
a
probability
distribution.
But
unlike
the
probability
distributions
we've
seen
thus
far,
the
"variable"
of
interest
here
is
not
an
individual
outcome,
but
rather
is
a
sample
statistic.
The
probability
distribution
for
a
sample
statistic
is
called
a
"Sampling
Distribution"
1
CHARACTERISTICS OF
THE
"SAMPLING DISTRIBUTION
OF
THE
MEAN"
Has
a
mean
of
Has
a
standard
deviation
of
Has
a
shape/pattern,
according
to
the
Central
Limit
Theorem
NOTE
: wh
en
a
population
is
described
by
a
probability
distribution,
the
mean
of
the
population
is
also
referred
to
as
the
"expected
value",
defined
as
I
I
I
I
2::
X
P(X)
So
the
"expected
value"
of
our
derived
population
of
sample
mean
s
here
is
found
by:
E(Xbar)=l.5(.1)+2.0(.1)+2.5(.2)+3.0(.2)+3.5(.2)+4.0(.1)+4.5(.1)
i .
e.
E(Xbar)
=
p(Xbar)
=
3.0
And
the
mean
of
our
original
population
here
was:
p
=
1 + 2 + 3 + 4 + 5
p
=
3
5
And
this
equality
isn't
coinc
iden
tal
here,
nor
a
quirk
of
the
small
population
or
sample
size
.