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.

AWhat is the distribution for the weights of one 25-pound lifting weight? What is the mean and standard deivation?

BWhat is the distribution for the mean weight of 100 25-pound lifting weights?

CFind the probability that the mean actual weight for the 100 weights is less than 24.9.