CK-12 Probability and Statistics - Advanced

7.2. The z-Score and the Central Limit Theorem http://www.ck12.org

c)

σx ̄=

σ

√

n

σx ̄=

4. 3

√

40

σx ̄= 0. 68

A sample with a sample size of 40 is taken from a known population whereμ=25 andσ=4. The following chart
displays the collected data:

24 23 30 17 24 22 23 21 29 25

26 25 29 28 29 29 32 22 27 28

24 32 21 29 30 18 21 24 30 24

25 26 25 27 26 25 27 24 24 25

a) What is the population mean?

b) Determine the sample mean using technology.

c) What is the population standard deviation?

d) Using technology, determine the sample standard deviation.

e) If an infinite number of samples of size 40 were collected from this population, what would be the value of the
sample means?

f) If an infinite number of samples of size 40 were collected from this population, what would be the value of the
standard deviation of the sample means?

Solution:

a)μ=25 The population mean of 25 was given in the question.

b) ̄x= 25 .5 The sample mean is 25.5 and is determined by using 1−Vars Staton the TI-83.

c)σ=4 The population standard deviation of 4 was given in the question.

d)Sx= 3 .47 The sample standard deviation is 3.47 and is determined by using 1−Vars Staton the TI-83.

e)μx ̄=25 A property of the Central Limit Theorem.

f)

σx ̄=

σ

√

n

σx ̄=

4

√

40

σx ̄= 0 .63 Central Limit Theorem

Lesson Summary

For approximately normal distributions, the mean and the standard deviation are used as the measure of center and
spread. If these two values are known, thez-scores and the calculator can be used to find the percentage of values in
any interval.