7.1 The Central Limit Theorem for Sample
Means
The Central Limit Theorem for Sample Means:
X
–
~ N
⎛
⎝
µ x
– ,
σ
n
⎞
⎠
Z =
X
–
− µ
X
–
σ
X
– =
X
–
- µ
σ / n
The Mean X
–
: µ x
–
Central Limit Theorem for Sample Means z-score
z =
x
–
− µ x
–
⎛
⎝
σ
n
⎞
⎠
Standard Error of the Mean (Standard Deviation ( X
–
)):
σ
n
Finite Population Correction Factor for the sampling
distribution of means: Z =
x
¯
− µ
σ
n
*
N − n
N − 1
Finite Population Correction Factor for the sampling
distribution of proportions: σp' =
p(1 − p)
n
×
N − n
N − 1
PRACTICE
7.2 Using the Central Limit Theorem
Use the following information to answer the next ten exercises: A manufacturer produces 25-pound lifting weights. The
lowest actual weight is 24 pounds, and the highest is 26 pounds. Each weight is equally likely so the distribution of weights
is uniform. A sample of 100 weights is taken.
1.
a. What is the distribution for the weights of one 25-pound lifting weight? What is the mean and standard deivation?
b. What is the distribution for the mean weight of 100 25-pound lifting weights?
c. Find the probability that the mean actual weight for the 100 weights is less than 24.9.