Chapter 7: Probability and Samples: The Distribution of Sample Means -…
Chapter 7: Probability and Samples: The Distribution of Sample Means
Distribution of sample means, allows us to find the exact probability of obtaining a specific sample mean from a specific population.
This distribution describes the entire set of all the possible sample means for any sized sample.
Since we can describe the entire set, we can find probabilities associated with specific sample means.
Sampling error is the natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter.
The difficulty of working with samples is that a sample provides an incomplete picture of the population.
In addition, any statistics that are computed for the sample will not be identical to the corresponding parameters for the entire population.
Samples are variable; they are not all the same.
Two separate samples from the same population, would yield samples that will be different.
In many cases, it is possible to obtain thousands of different samples from one population.
A sampling distribution is a distribution of statistics obtained by selecting all the possible samples of a specific size from a population.
Often is called the sampling distribution of M.
To construct the distribution of sample means, you first select a random sample of a specific size (n) from a population, calculate the sample mean, and place the sample mean in a frequency distribution.
After select another random sample with the same number of scores. Again, you calculate the sample mean and add it to your distribution.
Characteristics of the Distribution of Sample Means are as follows:
Sample means should tend to form a normal-shaped distribution.
The sample means should pile up around the population mean. Samples are not expected to be perfect but they are representative of the population.
Due to this, most of the sample means should be relatively close to the population mean.
Most samples should have means close to μ, and it should be relatively rare to find sample means that are substantially different from μ. Due to this, the sample means pile up in the center of the distribution (around μ) and the frequencies should taper off as the distance between M and μ increases.
Describes a normal-shaped distribution.
The larger the sample size, the closer the sample means should be to the population mean, μ. A large sample should be a better representative than a small sample. The sample means obtained with a large sample size should cluster relatively close to the population mean. Means obtained from small samples should be more widely scattered.
A mathematical proposition known as the central limit theorem provides a precise description of the distribution that would be obtained if you selected every possible sample, calculated every sample mean, and constructed the distribution of the sample mean.
Note that the central limit theorem describes the distribution of sample means by identifying the three basic characteristics that describe any distribution
Shape:The distribution of sample means is normal if either one of the following two conditions is satisfied:
(1) The population from which the samples are selected is normal.
(2) The size of the samples is relatively large (around n= or more).
Central-Tendency: The mean of the distribution of sample means is identical to the mean of the population from which the samples are selected. The mean of the distribution of sample means is called the expected value of M.
Variability: The standard deviation of the distribution of sample means is called the standard error of M and is defined by the formula
Serves as a cornerstone for much of inferential statistics