CK-12 Probability and Statistics - Advanced

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

rewritten using the symbols from the chart above. The formulas=

√

P·Q

n

can be expressed as the quotient of two

radical expressionss=

√

√P·Q

n

. The square root of the product of the parametersPandQis actually the standard

deviation of the population(σ). When this value is divided by square root of the sample size, the result is the

standard error(s), also known as the standard deviation of the sampling distribution(Sx ̄). Therefores=

√

P·Q

n

can

be written asSx ̄= √σ
n
This frequency distribution only approximates the true sampling distribution of the sample

mean because a finite number of sample means were used. If, hypothetically, an infinite number of sample means
were used, the resulting distribution would be the desired sampling distribution and the following would be true:

σx ̄=

σ

√

n

The notationσx ̄reminds you that this is the standard deviation of the sample mean(x ̄)and not the standard deviation
(σ)of a single observation.

The Central Limit Theorem states the following:

• If samples of sizenare drawn at random from any population with a finite mean and standard deviation, then
  the sampling distribution of the sample mean(x ̄)approximates a normal distribution asnincreases.


• The mean of this sampling distribution approximates the population mean asnbecomes large:

μ≈μx ̄

• The standard deviation of the sample mean is approximately equivalent to the following

σx ̄=

σ

√

n

These properties of the sampling distribution of the mean can be applied to determining probabilities. The sampling
distribution of the sample mean can be assumed to be approximately normal, even if the population is not normally
distributed. Now that it has been clarified that the sampling distribution of the mean is approximately normal, let’s
see how these properties work. Suppose you wanted to answer the question, “What is the probability that a random
sample of 20 families in Canada will have an average of 1.5 pets or fewer?” where the mean of the population is 0. 8
and the standard deviation of the population is 1.2.

For the sampling distributionμx ̄=μ= 0 .8 andσx ̄= √σn=√^1.^2
20

≈ 0. 27

Using technology, a sketch of this problem is