Слайд 2The Distribution of Sample Means
In Chapter 7 we extend the concepts of
z-scores and probability to samples of more than one score.
We will compute z-scores and find probabilities for sample means.
To accomplish this task, the first requirement is that you must know about all the possible sample means, that is, the entire distribution of Ms.
Слайд 3The Distribution of Sample Means (cont.)
Once this distribution is identified, then
1. A
z-score can be computed for each sample mean. The z-score tells where the specific sample mean is located relative to all the other sample means.
2. The probability associated with a specific sample mean can be defined as a proportion of all the possible sample means.
Слайд 4The Distribution of Sample Means (cont.)
The distribution of sample means is defined
as the set of means from all the possible random samples of a specific size (n) selected from a specific population.
This distribution has well-defined (and predictable) characteristics that are specified in the Central Limit Theorem:
Слайд 5The Central Limit Theorem
1. The mean of the distribution of sample means is
called the Expected Value of M and is always equal to the population mean μ.
The standard deviation of the distribution of sample means is called the Standard Error of M and is computed by
σ σ2
σM = ____ or σM = ____
√n n
3. The shape of the distribution of sample means tends to be normal. It is guaranteed to be normal if either a) the population from which the samples are obtained is normal, or b) the sample size is n = 30 or more.
Слайд 6The Distribution of Sample Means (cont.)
The concept of the distribution of sample
means and its characteristics should be intuitively reasonable:
1. You should realize that sample means are variable. If two (or more) samples are selected from the same population, the two samples probably will have different means.
Слайд 7The Distribution of Sample Means (cont.)
2. Although the samples will have different
means, you should expect the sample means to be close to the population mean. That is, the sample means should "pile up" around μ. Thus, the distribution of sample means tends to form a normal shape with an expected value of μ.
3. You should realize that an individual sample mean probably will not be identical to its population mean; that is, there will be some "error" between M and μ. Some sample means will be relatively close to μ and others will be relatively far away. The standard error provides a measure of the standard distance between M and μ.
Слайд 9z-Scores and Location within the Distribution of Sample Means
Within the distribution
of sample means, the location of each sample mean can be specified by a z-score,
M – μ
z = ─────
σM
Слайд 10z-Scores and Location within the Distribution of Sample Means (cont.)
As always, a
positive z-score indicates a sample mean that is greater than μ and a negative z-score corresponds to a sample mean that is smaller than μ.
The numerical value of the z-score indicates the distance between M and μ measured in terms of the standard error.
Слайд 12Probability and Sample Means
Because the distribution of sample means tends to be
normal, the z-score value obtained for a sample mean can be used with the unit normal table to obtain probabilities.
The procedures for computing z-scores and finding probabilities for sample means are essentially the same as we used for individual scores
Слайд 13Probability and Sample Means (cont.)
However, when you are using sample means, you
must remember to consider the sample size (n) and compute the standard error (σM) before you start any other computations.
Also, you must be sure that the distribution of sample means satisfies at least one of the criteria for normal shape before you can use the unit normal table.
Слайд 15The Standard Error of M
The standard error of M is defined as
the standard deviation of the distribution of sample means and measures the standard distance between a sample mean and the population mean.
Thus, the Standard Error of M provides a measure of how accurately, on average, a sample mean represents its corresponding population mean.
Слайд 17The Standard Error of M (cont.)
The magnitude of the standard error is
determined by two factors: σ and n.
The population standard deviation, σ, measures the standard distance between a single score (X) and the population mean.
Thus, the standard deviation provides a measure of the "error" that is expected for the smallest possible sample, when n = 1.
Слайд 18The Standard Error of M (cont.)
As the sample size is increased, it
is reasonable to expect that the error should decrease.
The larger the sample, the more accurately it should represent its population.