Normal Distribution and Estimation

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Normal Distribution and Estimation

• Iman Adibi
• Student's Research Committee
• (i_adibi _at_med.mui.ac.ir)

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objectives

• Concept of inference and estimation
• Concepts in probability
• Probability distribution
• Normal distribution
• Z distribution
• Standard error and confidence interval
• Estimation of proportion

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What is research ?
Scientific question
population
x
sample
Scientific answer
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Sources of Error

• Errors from biased sampling
• The study systematically favors certain
  outcomes
• Voluntary response
• Non-response
• Convenience sampling
• Solution Random sampling

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Sources of Error

• Errors from (random) sampling
• Caused by chance occurrence
• Get a bad sample because of bad luck (by
  bad we mean not representative)
• Can be controlled by taking a larger sample

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• INFERENCE
• Methodologies that allow us to draw conclusions
  about population parameters from sample
  statistics
• There are two procedures for making inferences
• Estimation.
• Hypotheses testing

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Concepts of Estimation

• The objective of estimation is to determine the
  value of a population parameter on the basis of a
  sample statistic.
• There are two types of estimators
• Point Estimator
• Interval estimator

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Point Estimator
A point estimator draws inference about a
population by estimating the value of an unknown
population parameter using a single value or a
point.
Parameter
Population distribution
?
Sample distribution
Point estimator
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Interval Estimator

• An interval estimator draws inferences about a
  population by estimating the value of an unknown
  population parameter using an interval.

Interval estimator
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Estimators desirable characteristics

• Unbiasedness An unbiased estimator is one whose
  expected value is equal to the parameter it
  estimates.
• Consistency An unbiased estimator is said to be
  consistent if the difference between the
  estimator and the parameter grows smaller as the
  sample size increases.
• Relative efficiency For two unbiased estimators,
  the one with a smaller variance is said to be
  relatively efficient.

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Confidence Interval Estimates
Confidence
Intervals

Mean
Proportion
?
?
Unknown
Known
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Concepts in probability

• Addition Rule ( mutually exclusive events)
• P( A or B ) P(A) P(B)
• Addition Rule ( mutually exclusive events)
• P(A or B) P(A) P(B) P(A and B)

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Concepts in probability

• Multiplication Rule ( Independent Events)
• P(A and B) P(A) . P(B)
• Multiplication Rule (Dependent events)
• P(A and B) P(A) . P(B \ A)

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Concepts in probability

• Bayes theorem
• P(A and B) P(B and A)
• P(A) . P(B \ A) P(B) . P(A \ B)

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Concepts in probability

• Probability
• Odds
• Likelihood

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Random variablesand probability distribution

• Normal (Guassian ) distribution
• The Poisson distribution
• Binomial distribution

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Normal Distribution

• Mean gt median Right skewed
• Mean lt median left skewed
• Mean median mode symmetric

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The 68-95-99.7 Rule for theNormal Distribution

• 68 of the observations fall within one standard
  deviation of the mean
• 95 of the observations fall within two
  standard deviations of the mean
• 99.7 of the observations fall within three
  standard deviations of the mean
• When applied to real data, these estimates are
  considered approximate!

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Three Common Areas Under the Curve
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The Relationship Between Z and X
? 100 ? 15
P(X)lt130
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Theoretical normal distribution with standard
deviations
-3?
-2?
-?
?
?
2?
3?
Z scores
-3
-2
-1
1
2
3
0
Upper tail .1587 .02288
.0013 Two-tailed .3173 .0455 .0027
Probability
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The standard normal (z) distribution

• A normal distribution
• Mean 0
• Standard deviation 1

X - m
Z
s
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• What is the z score for 0.05 probability?
• (one-tailed test) 1.645
• What is the z score for 0.05 probability? (two
  tailed test) 1.96
• What is the z score for 0.01?
• (one-tail test) 2.326
• What is the z score for 0.01 probability?
• (two tailed test) 2.576

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Sample distribution

• How to make a sample distribution
• Sampling distribution of mean
• The Central Limit Theorem
• Given a population with mean m standard deviation
  , the sampling distribution of the mean based on
  repeated random samples of size n has the
  following properties

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The Central Limit Theorem

• The mean of the sampling distribution or the mean
  of the means is equal to population mean based
  on the individual observations

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The Central Limit Theorem

• 2. The standard deviation in the sampling
  distribution of the mean is called standard error
  of mean . This quantity called the standard error
  of the mean.

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The Central Limit Theorem

• 3. If the distribution in the population is
  normal then the sampling distribution of the mean
  is also normal
• More importantly for sufficiently large sample
  sizes the sampling distribution of mean is
  approximately normally distributed regardless of
  the shape of the original population distribution
  (n gt 30 )

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Estimating the Population Mean When the
Population Variance Is Unknown

• Recall that when s is known, the statistic z is
  normally distributed
• if the sample is drawn from a normal
  population, or
• if the population is not normal but the sample is
  sufficiently large.

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Estimating the Population Mean When the
Population Variance Is Unknown

• When s is unknown, we use its point estimator s,
  and the Z statistic is replaced then by the
  t-statistic

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.025
Normal distribution of
.025
.025
m
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1 - a
Upper confidence limit
Lower confidence limit
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• Amazing facts about confidence intervals(for
  normally distributed statistics)
• To halve the interval, you have to quadruple
  sample size.
• A 99 interval is 1.3 times wider than a 95
  interval.You need 1.7 times the sample size for
  the same width.
• A 90 interval is 0.8 of the width of a 95
  interval.You need 0.7 times the sample size for
  the same width.

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Estimating the population mean when the
population variance is unknown

• Confidence interval estimator of m when s is
  unknown

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Students t Distribution
Standard Normal
Bell-Shaped Symmetric Fatter Tails
t (df 13)
t (df 5)
Z
t
0
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Z
t
s
Where
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Confidence Interval Estimate for Proportion

• Assumptions
• Two categorical outcomes
• Population follows binomial distribution
• Normal approximation can be used if
  and
• Confidence interval estimate

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objectives

• Concept of inference and estimation
• Concepts in probability
• Probability distribution
• Normal distribution
• Z distribution
• Standard error and confidence interval
• Estimation of proportion

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