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Chapter 5

Sociology 2205A/B Chapter Notes - Chapter 5: Simple Random Sample, Statistical Inference, Sampling Distribution


Department
Sociology
Course Code
SOC 2205A/B
Professor
Anna Zajacova
Chapter
5

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SOC 2205 CHAPTER 5
Introduction to Inference Sampling Distributions
5.2 Probability Sampling
The goal of probability sampling is to select cases so that the final sample is
representative of the population from which it was drawn
o It is representative if it reproduces the important characteristics of the
population
o It is very much like the population only smaller
We can maximize the chances of a representative sample by following the principle
of EPSEM (the Equal Probability of SElection Method)
EPSEM equal probability of selection method. A technique for selecting samples
in which every element or case in the population has an equal probability of being
selected for the sample
EPSEM sampling technique produces a simple random sample (a sample drawn
from a population so that every case has an equal chance of being included)
Summary: the purpose of inferential statistics is to acquire knowledge about
populations, based on information derived from samples of that population
5.3 The Sampling Distribution
Sampling distribution the theoretical, probabilistic distribution of a statistic for
all possible samples of a certain sample size (n)
o Includes statistics that represent every possible sample from the population
The characteristics in sampling distribution are based on the laws of probability not
on empirical information
Three distinct distributions are involved in every application of inferential statistics:
1. The sample distribution, which is empirical (i.e., it exists in reality) and
known in the sense that the shape, central tendency, and dispersion of may
variable can be ascertained for the sample
2. The population distribution, which, while empirical, is unknown
3. The sampling distribution, which is non-empirical or theoretical - the shape,
central tendency, and dispersion of the distribution can be deduced, and
therefore, the distribution can be adequately characterized
The Relationships between the Sample, Sampling Distribution, and Population
Example: suppose we know that the true mean age of a population is 30
o Most of the sample means will also be approximately 30 and the sampling
distribution of these sample means should peak at 30
o The distribution should slope to the base as we get farther away from the
population value
o Means of 29 or 31 should be common; means of 20 or 40 should be rare
o The distribution is roughly symmetrical
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o Recall, that on any normal curve, cases close to the mean (within +/-1
standard deviation) are common and those (beyond +/-3 standard
deviations) are rare
A Sampling Distribution of Sample Means
The shape of the sampling distribution, information about central tendency and
dispersion are stated in two theorems:
1) if repeated random samples of size n are drawn from a normal population with
mean and standard deviation, then the sampling distribution of sample means will be
normal with a mean and a standard deviation of: standard deviation over n square root
o Translate if we begin with a trait that is normally distributed across a
population (IQ, height, weight) and take an infinite number of equally sized
random samples from that population, then sampling distribution of the sample
means will be normal. If it known that the variable is distributed normally in the
population, it can be assumed that the sampling distribution will be normal
o The mean of the sampling distribution will be exactly the same value as the mean
of the population
e.g. if we know that the mean IQ of the entire population is 100, then we
know that the mean of any sampling distribution of sample means IQs will
also be 100
o The theorem states that the standard deviation of the sampling distribution
(also called the standard error), will be equal to the standard deviation of the
population divided by the square root of
n
o This theorem requires a normal population distribution
2) Central Limit Theorem: if repeated random samples of size n are drawn from any
population, with mean and standard deviation, then, as n becomes large, the
sampling distribution of sample means will approach normality, with mean and
standard deviation standard deviation of population divided by the square root
of n
o Translation: for any trait or variable, even those that are not normally
distributed in the population, as sample size grows larger, the sampling
distribution of sample means will become normal in shape
o Importance of the theorem is that it removes the constraint of normality in
the population
o Utilized when the distribution of the variable in question is unknown or is
known to not be normal in shape (i.e., income, which always has a positive
skew)
5.4 Constructing the Sampling Distribution
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