Central limit theorem

Theorem: Whatever the population, the distribution of is approximately

normal when n is large. In random sampling from the population with mean and standard deviation , when n is large, the distribution of is approximately normal with mean and standard deviation . Consequently,

is approximately

Remark:

The sample size is usually considered to be large if .

The central limit theorem states that the distribution of is normally distributed and can be used to compute a random variable, Z, with mean 0 and variance 1:

.

Example:

The amount of telephone bills for all households in a large city have a distribution that is skewed to the right with a mean of $30 and a standard deviation equal to $7. Calculate the mean and standard deviation of and describe the shape of its sampling distribution when the sample size is

a) n = 35; b) n =70

Solution:

Although the population distribution is not normal, in each case the sample size is large .

Hence the central limit theorem can be applied.

a) Let be the mean value of telephone bills paid by a sample of 35 households. Then the sampling distribution of is approximately normal with

and .

Figure 5.3 shows the population distribution and sampling distribution of .

b) Let be the mean value of telephone bills paid by a sample of 70 households. Then the sampling distribution of is approximately normal with

and .

Figure 5.4 shows the population distribution and sampling distribution of .

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Example:

Consider a population with mean 75 and standard deviation of 11.

a) If a random sample of size 64 is selected, what is the probability that the sample mean will be between 73 and 78?

b) If a random sample of size 80 is selected, what is the probability that the sample mean will be between 68 and 83?

Solution:

We have and Since n =64 is large, the central limit theorem tells us that the distribution of is approximately normal.

a) To calculate we convert to the standardized variable

.

The Z –values corresponding to 73 and 78 are

For 73:

For 78:

Consequently,

b) We now have n =80.

.

Example:

The prices of all houses in a large city have a probability distribution with a mean of $80 000 and a standard deviation of $15 000. Let be the mean price of a sample of 200 houses selected from this city.

a) What is the probability that the mean price obtained from this sample will be within $2 000 of the population mean?

b) What is the probability that the mean price obtained from this sample will be more than the population mean by $1 500 or more?

Solution:

The sampling distribution of is approximately normal because the sample size is large (n >30).

a) We need to find the probability

.

b) The probability that the mean price obtained from the sample of 200 houses will be more than the population mean by $1 500 or more is written as

.

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