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Chapter 6

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

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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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