Statistics with Economics and Business Applications

1
Statistics with Economics and Business
Applications
Chapter 7 Estimation of Means and
Proportions Point Estimation, Interval
Estimation/Confidence Interval
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Review

• I. Whats in last lecture?
• Random Sample, Central Limit Theorem
  Chapter 6
• II. What's in this lecture?
• Point Estimation
  Interval Estimation/Confidence Interval
• Read Chapter 7

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Probability vs Statistics Reasoning

• In the last Chapter we looked at sampling
  staring with a population, we imagined taking
  many samples and investigated how sample
  statistics were distributed (sampling
  distribution)
• In this Chapter, we do the reverse given one
  sample, we ask what was the random system that
  generated its statistics
• This shift our mode of thinking from deductive
  reasoning to induction

Probability/deduction
Population
Sample
Statistics/induction
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Probability vs Statistics Reasoning

• Deductive reasoning from a hypothesis to a
  conclusion
• Inductive reasoning argues backward from a set
  of observations to a reasonable hypothesis
• In many ways, science, including statistics, is
  like detective work. Beginning with a set of
  observations, we ask what can be said about the
  system that generated them

Data! Data! Data! I can't make bricks without
clay. Sherlock Holmes.
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Parameters

• Populations are described by their probability
  distributions and/or parameters.
• For quantitative populations, the location and
  shape are described by m and s.
• Binomial populations are determined by a single
  parameter, p.
• If the values of parameters are unknown, we make
  inferences about them using sample information.

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Types of Inference

• Estimation (Chapter 7)
• Estimating or predicting the value of the
  parameter
• What is (are) the most likely values of m or
  p?
• Hypothesis Testing (Chapter 8)
• Deciding about the value of a parameter based on
  some preconceived idea.
• Did the sample come from a population with m 5
  or p .2?

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Types of Inference

• Examples
• A consumer wants to estimate the average price of
  similar homes in her city before putting her home
  on the market.

Estimation Estimate m, the average home price.

• A manufacturer wants to know if a new type of
  steel is more resistant to high temperatures than
  an old type was.

Hypothesis test Is the new average resistance,
mN greater than the old average resistance, mO?
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Types of Inference

• Whether you are estimating parameters or testing
  hypotheses, statistical methods are important
  because they provide
• Methods for making the inference
• A numerical measure of the goodness or
  reliability of the inference

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Definitions

• An estimator is a rule, usually a formula,
  that tells you how to calculate the estimate
  based on the sample.
• Point estimation A single number is calculated
  to estimate the parameter.
• Interval estimation/Confidence Interval Two
  numbers are calculated to create an interval
  within which the parameter is expected to lie. It
  is constructed so that, with a chosen degree of
  confidence, the true unknown parameter will be
  captured inside the interval.

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Point Estimator of Population Mean
An point estimate of population mean,
, is the
sample mean
A sample of weights of 34 male freshman students
was obtained. 185 161 174 175 202 178 202 139 177
170 151 176 197 214 283 184 189 168 188 170 207 18
0 167 177 166 231 176 184 179 155 148 180 194 176
If one wanted to estimate the true mean of all
male freshman students, you might use the sample
mean as a point estimate for the true mean.
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Point Estimation of Population Proportion
An point estimate of population mean, p, is the
sample proportion
, where x is the number of
successes in the sample.

• A sample of 200 students at a large university is
  selected to estimate the proportion of students
  that wear contact lens. In this sample 47 wear
  contact lens.

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Properties of Point Estimators

• Since an estimator is calculated from sample
  values, it varies from sample to sample according
  to its sampling distribution.
• An estimator is unbiased if the mean of its
  sampling distribution equals the parameter of
  interest. It does not systematically overestimate
  or underestimate the target parameter.
• Both sample mean and sample proportion are
  unbiased estimators of population mean and
  proportion. The following sample variance is an
  unbiase estimator of population variance.

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Properties of Point Estimators

• Of all the unbiased estimators, we prefer the
  estimator whose sampling distribution has the
  smallest spread or variability.

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Interval Estimators/Confidence Intervals

• Confidence intervals depend on sampling
  distributions
• The shape of sampling distributions depend on
  sample sizes
• We will learn different methods for large and
  small sample sizes
• For large sample sizes, central limit theorem
  applies which allow us to use normal
  distributions
• For small sample sizes, we need to learn a new
  distribution

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Key Concepts

• I. Types of Estimators
• 1. Point estimator a single number is
  calculated to estimate the population parameter.
• 2. Interval estimator/confidence interval range
  of values which is likely to include an unknown
  population parameter.
• II. Properties of Good Estimators
• 1. Unbiased the average value of the estimator
  equals the parameter to be estimated.
• 2. Minimum variance of all the unbiased
  estimators, the best estimator has a sampling
  distribution with the smallest standard error.