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Chapter 6

Sociology 2205A/B Chapter Notes - Chapter 6: Interval Estimation, Bias Of An Estimator, Statistical Inference


Department
Sociology
Course Code
SOC 2205A/B
Professor
Anna Zajacova
Chapter
6

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SOC 2205 Chapter 6
Estimation Procedures
6.1 Introduction
The objection of this branch of inferential statistics is to estimate population values
or parameters from statistics computed from samples
Most common applications: public opinion polls and election projections
There are two kinds of estimation procedures:
o Point estimation a sample statistic that is used to estimate a population
value; e.g. report that states that 50% of a sample randomly selected drivers
are driving less because of high gas prices
o Confidence intervals consist of a range of values (an interval) instead of a

report driving less than usual due to high 
o The size of the confidence interval, in this example it is 6% (the percentage
difference between 47% and 53%, is called the margin of error, or sampling
error
6.2 Bias and Efficiency
Estimators can be selected according to two criteria: bias and efficiency
Bias a criterion used to select sample statistic for estimation procedures. It is
unbiased if the mean of its sampling distribution is equal to the population value of
interest
Efficiency the extent to which sample outcomes are clustered around the mean of
the sampling distribution
Bias
In chapter 5, the mean of the sampling distribution of sample means is the same as
the population mean
Sample proportions are also unbiased, if we calculate sample proportions from
repeated random samples of size n, the sampling proportions will have a mean
equal to the population proportion
However, other statistics are biased (i.e., have sampling distributions with means
not equal to the population value)
o In particular, the sample standard deviation (s) is a biased estimator of the
population standard deviation. There is less dispersion in a sample than in a
population and, as a consequence, s will underestimate the pop. S.d. The
sample standard deviation can be corrected for this bias
Example: we wish to estimate the average household income of a community;
random sample of 500 households are taken (n=500), sample mean is $75,000,
what is the value of the population mean
o Because n is large, we known that the sampling distribution is normal and
that its mean is equal to the population mean
o We also know that all normal curves contain about 68% of the cases within
+/-1Z, 95% cases are +/-2Z, and more than 99% are within +/-3Z of the
mean
We are discussing the sampling distribution
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The probabilities are very good that our sample mean of $75,000 is within +/-1Z,
excellent that it is within +/--
mean of the sampling distribution
Less than 1% of cases a sample mean will be more than +/-
mean
Efficiency
*the standard deviation of the sampling distribution of sample means, or the
standard error of the mean, is equal to the population standard deviation divided by
the square root of n
as sample size increases, the standard deviation of the sampling distribution will
decrease
The smaller the standard deviation of a sampling distribution, the greater the
clustering and the higher the efficiency
= $5,000
A Sampling Distribution with and n=100 and
) = $500.00
A Sampling Distribution with and n=1,000 and
) = 158.11
6.3 Estimation Procedures: Introduction
1. Draw an EPSEM sample
2. Calculate either a proportion or a mean
3. Estimate that the population parameter is the same as the sample statistic
The larger the sample, the greater the efficiency and the more likely that the estimator
is approx. The same as the population value
Constructing an Interval Estimate
i) Decide on the risk you are willing to take of being wrong
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