Samples and Populations
z-scores and probabilities are limited to situations in which the sample consists of a single score. Most research involves much larger samples and so, in these situations the sample mean, rather than a single score, is used to answer questions about the
Sampling Error
The natural discrepancy, or amount of error, between a sample statistic and it's corresponding population parameter.
The Distribution of Sample Means
The collection of sample means for all of the possible random samples of a particular size (n) that can be obtained from a population. It contains all of the possible samples and so it is necessary to have all of the possible values to compute probabiliti
Sampling Distribution
A distribution of statistics obtained by selecting all of the possible samples of a specific size from a population. The distribution of sample means is an example of a sampling distribution.
General Characteristics of Sample Distribution
1. The sample means should pile up around the population mean.
2. The pile of sample means should tend to form a normal shaped distribution and the frequencies should taper off as the distances between M and ? increases.
3. In general, the larger the samp
Central Limit Theorem
For any population with mean ? and standard deviation ?, the distribution of sample means for sample size n will have the mean ? and standard deviation ?/n^(1/2) and will approach a normal distribution as n approaches infinity. This theorem describes the
The Shape of the Distribution of Sample Means
The distribution of sample means tends to be a normal distribution. It is perfectly normal if other of the two following conditions are satisfied:
1. The population from which the samples are selected is a normal distribution.
2. The number of scores (n)
The Mean of the Distribution of Sample Means: The Expected Value of M
The average value of all the sample means is exactly equal to the value of the population mean. The formal statement of this phenomenon is that the mean of the distribution of sample means always is identical to the population mean. This mean value is cal
Expected Value of M
The mean of the distribution is equal to the mean of the populations of scores, ?, and is called the
expected value of M
*.
The Standard Error of M
Describes the standard deviation of the distribution of sample means. It provides a measure of how much difference is expected from one sample to another. When the standard error is small, then all of the sample means are close together and have similar v
The Law of Large Numbers
States that the larger the sample size (n) , the more probable it is that the sample mean is close to the population mean. In general, as the sample size increases, the error between the sample mean and the population mean should decrease.
The Magnitude of the Standard Error
The magnitude of the standard error is determined by two factors: the size of the sample and the standard deviation of the population from which the sample is selected.
The Magnitude of the Standard Error: Sample Size
The size of the sample should influence how accurately the sample represents the population. A large sample is more accurate than a small sample.
The Magnitude of the Standard Error: Population Standard Deviation
There is an inverse relationship between the sample size and the standard error: bigger samples have smaller error and smaller samples have bigger error. The formula for standard error expresses this relationship between standard deviation and sample size