The Distribution of the Sample Mean
Example 1:
A bottling company uses a filling machine to fill plastic bottles with a popular cola. The
bottles are supposed to contain 300 ml. In fact, the contents vary according to a normal
distribution with a J.1 = 298 ml and (Y = 3 ml.
What is the probability that a randomly selected bottle contains < 295
ml?
295 -298
P(Bottle x < 295) = P(Z<
)= -1.00 = .1587
3
Thus, there is nearly a 16% probability that just 1 randomly selected
bottle contains < 295 ml.
Example 2:
What is the probability that the average contents of 6 randomly selected
bottles is < 295?
P(x < 295) = P(z
-3
295-298
.
(
Ch angmg x to z < 3//6
)-=-2.45
1.22
x-J.1
z < -2.45
z=--
1~
=.0071
Thus, there is < 1% probability that 6 randomly selected bottles
contain < 295 ml.
Note: A larger sample equals a smaller probability.
Example 3:
J.1 = 8000
(Y
= 3200
P( x> 9000) = P(z > 9000 - 8000 = 1000
3200/
320%
J64
n=64
z> 2.5 = .0062
Example 4:
1.
(5,
~ ~ ~ SD of distribution
IT(l- IT)
n
Assuming the proportion of red M&Ms is actually 25%. What is the probability
that the proportion of red M&Ms for a sample of size 30 is 22% or less?
P(p<.22)
=
P( z <
.22 - .25
.25(1- .25)
30
= p(z < - .03
)p(z < -.38) = .3520
.08
There is a 35% probability that the proportion of red M&Ms for a sample of 30 is ::::
22%.
-