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Chapter 7 - Probability and Samples: The Distribution of Sample Means -…
Chapter 7 - Probability and Samples: The Distribution of Sample Means
7-1 Samples, Populations, and the Distribution of Sample Means
sampling error
- the natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter
a z-score value near zero indicates a central, representative sample
a z-value beyond +2.00 or -2.00 indicates an extreme sample
the difficulty of working with samples is that a sample provides an incomplete picture of the population
any statistics that are computed for the sample will not be identical to the corresponding parameters for the entire population
samples are variable
The Distribution of Sample Means
Two separate samples will be different even if taken from the same population - different individuals, scores, means, etc.
Possible to obtain many thousands, or even millions, of different samples from one population
the huge set of possible samples forms a relatively simple and orderly pattern that makes it possible to predict the characteristics of a sample with some accuracy
Distribution of sample means
- the collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population
example of a sampling distribution
called the sampling distribution of M
Sampling Distribution
- a distribution of statistics obtained by selecting all the possible samples of a specific size from a population
Characteristics of the Distribution of Sample Means
The general characteristics of the distribution
The sample means should pile up around the population mean. (because samples are representative of the population, so should be relatively close to population mean)
The pile of sample means should tend to form a normal-shaped distribution. Most samples should have means close to μ and therefore, sample means should pile up in the center of the distribution and frequencies should taper off as the distance between M and μ increases.
The larger the sample size, the closer the sample means should be to the population mean, μ. The sample means obtained with a large sample size should cluster relatively close to the population mean
7-2 Shape, Central Tendency, and Variability for the Distribution of Sample Means
The Central Limit Theorem
A mathematical proposition that states: For any population with mean μ and standard deviation σ, the distribution of sample means for sample size n will have a mean of μ and a standard deviation of σ/√n and will approach a normal distribution as n approaches infinity.
The Shape of the Distribution of Sample Means
The distribution of sample means tends to be a normal distribution and is almost perfectly normal if:
The population fro which the samples are selected is a normal distribution.
The number of scores (n) in each sample is relatively large, around 30 or more.
As n gets larger, the distribution of sample means will closely approximate a normal distribution
The Mean of the Distribution of Sample Means: The Expected Value of M
The mean of the distribution of sample means always is identical to the population mean
The
expected value of M
is the mean of the distribution of sample means; it is equal to the mean of the population of scores, μ
the average value of M (the average value of all the sample means) is equal to μ
The sample mean is an example of an unbiased statistic, which means that on average the sample statistic produces a value that is exactly equal to the corresponding population parameter
The Standard Error of M
Standard error of M
- standard deviation for the distribution of sample means; σM
Two purposes for the distribution of sample means:
The standard error describes the distribution of sample means.
small standard error, all sample means are closer together and have similar values
larger standard error, sample means are scattered over a wide range
Standard error measures how well an individual sample mean represents the entire distribution.
provides measure of how much distance is reasonable to expect between a sample mean and the overall mean for the distribution of sample means
provides a measure of how much distance to expect between a sample mean (M) and the population mean (μ)
Law of large numbers
- rule that states that the larger the sample size (n), the more probable it is that the sample mean will be close to the population mean
when n = 1, the standard error = σM is identical to the standard deviation = σ
Defining the Standard Error in Terms of Variance
Three Different Distributions
Original population of scores that contains the scores for thousands or millions of individual people, and it has its own shape, mean, and standard deviation.
Sample selected from the population, consisting of a small set of scores for people who have been selected to represent the entire population
Distribution of sample means - a theoretical distribution consisting of the sample means obtained from all the possible random samples of a specific size
7-3 z-Scores and Probability for Sample Means
The primary use for the distribution of sample means is to find the probability of selecting a sample with a specific mean
A z-Score for Sample Means
Finding the location for a sample mean (M) rather than a score (X).
the standard deviation for the distibution of sample means is the standard error, σM
z-score for a sample mean can be difined as a signed number that identifies the location of the sample mean in the distribution of sample means so that:
the sign tells whether the sample mean is located above (+) or below (-) the mean for the distribution
the number tells the distance between the sample mean and μ in terms of the number of standard errors
z-score formula for locating a sample mean
7-4 More about Standard Error
Whenever you are working with a sample mean, you must use the standard error.
Sampling Error and Standard Error
An example of a typical distribution of sample means.
Each bar represents the frequencies for different sample means.
A curve for the normal distribution is superimposed on the histogram.
The expected value of the distribution of sample means equals the population, μ
Sampling Error - a sample typically will not provide a perfectly accurate representation of its population.
Standard Error - provides a way to measure the 'average," or standard, distance between a sample mean and the population menu
