Ch. 7: The Sampling Distribution of the Sample Mean

Sampling Distribution of the Sample Mean for a Normally Distributed Variable

Suppose that s variable x of a population is normally distributed with mean μ and standard deviation σ. Then, for samples of size n, the variable x̄ is also normally distributed and has mean μ and standard deviation σ/ sq root of n

Standard Deviation of the Sample Mean

is also called the standard error (SE) of the sample mean

Sample Size and Sampling Error

the larger the sample size, the smaller the sampling error tends to be in estimating a population mean, μ, by a sample mean, x̄

Sampling Distribution of Sample Mean

•probability distribution of all sample means computed from all possible random samples of a specific size n taken from the same population with replacement

Sampling Error

-difference between the sample measure and the corresponding population measure, due to the fact that the sample is not a perfect presentation of the population -discrepancy between the sample and the population

Central Limit Theorem

For a relatively large sample size, the variable x̄ is approximately normally distributed, regardless of the distribution of the variable under consideration. The approximation becomes better with increasing sample size. -If the original value is normally distributed, then the sample mean will be normally distributed. -If the original value is not normally distributed, then a sample size of 30 or more is needed to approximate a sample mean distribution to a normal distribution. The larger the sample, the better approximation will be.

Mean of the Sample Mean

For samples of size n, the mean of the variable x̄ equals the mean of the variable under consideration -the mean of all possible sample means = the population mean

Standard Deviation of the Sample Mean

For samples of size n, the standard deviation of the variable x̄ equals the standard deviation of the variable under consideration divided by the square root of the sample size -the larger to sample size, the smaller the standard deviation of X bar -the smaller the standard deviation of X bar, the more closely the possible values of x bar (the possible sample means) cluster around the mean of x bar -the mean of x bar equals the population mean

Figure 7.3

Illustrates that the possible sample means cluster more closely around the population mean as the sample size increases. This result suggest that sampling error tends to be smaller for large samples than for small samples.

What generally happens to the sampling distribution of the sample mean as the sample size is decreased?

It becomes less tightly concentrated around the population mean.

What generally happens to the sampling error as the sample size is decreased?

It gets larger.

What is the sampling distribution of a statistic?

The distribution of all possible observations of the statistic for samples of a given size from a population