Sampling and Statistical Inference A Short Review

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Sampling and Statistical Inference A Short
Review

• MS 205
• Instructor Dr. Vince Yen

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Populations and Samples

• Population all items of interest for a
  particular study or investigation
• All married drivers in the U.S. over age 25
• All individuals who do not own a cell phone
• Sample a subset of a population
• Nielsen samples of TV viewers
• samples of invoices for audits
• Samples are used
• To reduce costs of data collection
• When a full census cannot be taken

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Concept of Population and Sample
Population distributio of X Normal(µ, s2) (mean,
variance)
Draw a sample of size n
X1
X2
. . .
Xn
Sample (mean, variance)
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Arithmetic Mean

• Population mean
• Sample mean
• Excel function AVERAGE(range)

Npopulation size
n sample size
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Variance

• Population
• variance
• Sample
• variance

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Standard Deviation

• Population
• Sample
• The standard deviation has the same units of
  measurement as the original data, unlike the
  variance

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Theoretical Properties of the Sampling
Distribution of the Mean

• Expected value of the sample mean is the
  population mean, m
• Variance of the sample mean is s2/n, where s2 is
  the variance of the population
• Standard deviation of the sample mean, called the
  standard error of the mean, is s/?n

x
µ
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Central Limit Theorem Sampling Distribution of
the Mean

• If the sample size is large enough (generally at
  least 30, but depends on the actual
  distribution), the sampling distribution of the
  mean is approximately normal, regardless of the
  distribution of the population.
• If the population is normal, then the sampling
  distribution of the mean is exactly normal for
  any n.

X ?(?, ?2)
n ? 30
X N(?, ?2)
X N(?, ?2/n)
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Standard Normal Distribution Probability Table
Lookup

• Transformation from N(m,s2) to N(0,1)
• Standard normal mean 0, variance 1, denoted
  as N(0,1)

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Statistical Analysis of Sample Data

• Estimation of population parameters
• Confidence intervals for population parameters
• Hypothesis testing to draw conclusions about
  population parameters or differences between them

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Theoretical Issues What Are Good Estimators?

• Unbiased estimator one for which the expected
  value equals the population parameter it is
  intended to estimate
• The sample variance is an unbiased estimator for
  the population variance

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Confidence Intervals

• Confidence interval (CI) an interval estimated
  that specifies the likelihood that the interval
  contains the true population parameter
• Level of confidence (1 a) the probability
  that the CI contains the true population
  parameter, usually expressed as a percentage
  (90, 95, 99 are most common).
• ? is called the level of significance

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Confidence Interval for the Mean s Known

• A 100(1 a) CI is ?x ? z?/2(?/?n)

z?/2 may be found from Normal ProbabilityTable or
using the Excel function NORMSINV(1-a/2)
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Confidence Interval for( Known)
Critical Value

• Assumptions
• Population standard deviation is known
• Population is normally distributed
• If population is not normal, use large sample
• Confidence Interval Estimate

Standard Error