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![Page 1: Introduction to Inferential Statistics. Introduction Researchers most often have a population that is too large to test, so have to draw a sample from.](https://reader031.fdocuments.in/reader031/viewer/2022032605/56649e7d5503460f94b7f5c6/html5/thumbnails/1.jpg)
Introduction to Inferential Statistics
![Page 2: Introduction to Inferential Statistics. Introduction Researchers most often have a population that is too large to test, so have to draw a sample from.](https://reader031.fdocuments.in/reader031/viewer/2022032605/56649e7d5503460f94b7f5c6/html5/thumbnails/2.jpg)
Introduction Researchers most often have a population that
is too large to test, so have to draw a sample from the population
Researchers collect a random sample from the population and generalize from the known characteristics of the sample to the unknown population
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Random Sampling To use inferential statistical techniques, it is required
that samples be randomly drawn from the population of interest Non-random samples can be used for exploratory research
Meaning that you are going to explore a topic to see what variables are important to the topic
But, the conclusions cannot be generalized to the population
Random sampling requires a precise process of selection
Need to remember that randomness is not representativeness Because a sample is random does not guarantee that it is
an exact representation of the population
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Random Sampling The probability is high that a randomly
selected sample will be representative The good thing about inferential statistics is
that they allow you to state the probability of this type of error very precisely
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Ways in Which Random Samples are Gathered
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Simple Random Sample For a simple random sample, each case and each
combination of cases in the population must have an equal probability of being chosen for the sample
This kind of sample is used when you have a complete list of all cases in the population
Most researchers use tables of random numbers to select cases
This would be extremely time consuming for a large sample, so another technique is used
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Systematic Sampling Needed for large populations Only the first case is randomly selected After that, every kth case is selected
Will choose the first case from the table of random numbers, then choose every 10th case, or however many you need to reach a sample of the size you want
Will divide your population size by the sample size to find the distance to the next score for your sample
This is not a random sample Need to make sure the list of the population is random
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Stratified Sample Proportional Stratified Sample
This is used if you want to guarantee a representation of certain categories of cases (the same percentage as in the population)
If you want to compare chemistry majors to criminology majors You first ask students their major Then you put them all in the same list sorted by
major, then begin your random sample from the list All majors will be included if your sample is large
enough, but each would be included in the proportion they are in the original population Will have more criminology majors
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Stratified Sample Disproportionate Stratified Sampling
If you need exactly the same number of students from each major You separate the students by major Then have to change your sampling fraction to account for
differences in number of cases If you need a sample of 50, and there are 100 zoology majors,
you would choose every other one to get exactly 50 cases
Problem with this is you cannot generalize directly to the population, since your sample will never be representative
The biggest problem in sampling is that there is no complete list of most populations
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Cluster Sample Used when there is no list of the members of
the population It involves random selection of geographical
units (states, neighborhoods, or blocks) And will test every case within the last
geographical units Not a random sample, so not as trustworthy,
but cheaper to do
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The Sampling Distribution
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Introduction Researchers have a great deal of information about
the sample distribution, but they know nothing about the population
It is the population that is of interest We do not want to know what 2,000 people think out of
the 100,000,000 or so adults in the U.S. What you would want to know about the distribution
of the population The shape of the distribution Some measure of central tendency Some measure of dispersion
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The Normal Curve You need to know the properties of the normal
curve, which are based on the laws of probability, to find out information about the population from the sample
To do this, you use a device known as the sampling distribution It bridges the gap between the sample and the population The sampling distribution is the central concept in
inferential statistics, so you need to understand the concept
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Three Distributions The sample distribution
This is empirical (observed) and known It is collected by researchers and used to learn about the
population The population distribution
It exists in reality, so is empirical, but it is unknown to the researcher
The sole purpose of inferential statistics is to make inferences (meaning draw conclusions) about the population distribution
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The Sampling Distribution This is nonempirical (theoretical)
Theoretical, since you only do one sample, and the sampling distribution is based on an infinite number of samples taken from that population
Laws of probability tell us much about this distribution
Theoretically, if you drew an infinite number of samples from a population Then you only computed the mean of each sample And you put the means on a graph to form a frequency
polygon
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The Sampling Distribution We know that each sample mean will be
slightly different Since each sample is not an exact representation
of the population We know that most of the sample means will
cluster around the true population value
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Two Theorems About the Sampling Distribution If repeated random samples of size N are
drawn from a normal population with mean µ and standard deviation σ, then the sampling distribution of sample means will be normal with a mean µ and a standard deviation of σ /the square root of N So the mean of the sampling distribution will
be the same as the mean of the population
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Theorems Since the samples are random, the means
should miss an equal number of times on either side of the population value Making the distribution symmetrical A normal curve with a bell shape So, we know about the shape of the sampling
distribution
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Dispersion of the Sampling Distribution We can also tell something about the
dispersion (specifically the standard deviation) of the sampling distribution
The formula for the standard deviation of the sampling distribution is represented by the symbol σ/the square root of N Which is the standard deviation of the population
divided by the square root of N
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Dispersion of the Sampling Distribution What this tells you is that in comparing a sampling
distribution with a population distribution, there will always be more variance in the population distribution
As the sample size gets larger, the variance of the sampling distribution will get smaller (N = the number in the sample)
The above theorem applies to populations that are normally distributed on a particular variable
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Central Limit Theorem This second theorem is needed if the
population distribution is not normal If repeated random samples of size N are
drawn from any population, with mean µ and standard deviation σ, then as N becomes large, the sampling distribution of sample means will approach normality, with mean µ and standard deviation σ/the square root of N
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Large Samples What, exactly, is meant by large
A good rule of thumb is that if N is 100 or more, the Central Limit Theorem applies, and you can assume that the sampling distribution is normal in shape