PARAMETRIC STATISTICAL INFERENCE

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PARAMETRIC STATISTICAL INFERENCE

• INFERENCE
• Methodologies that allow us to draw conclusions
  about population parameters from sample
  statistics
• TYPES OF INFERENCE
• Estimation
• Hypothesis testing
• Methods based on statistical relationships
  between samples and populations
• POINT ESTIMATION estimation of parameter from a
  sample statistic
• For the mean, standard deviation, etc..
• INTERVAL ESTIMATION using a sample to identify
  an interval within which the population parameter
  is thought to lie, with a certain probability

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ESTIMATION OF POPULATION MEAN

• Sample mean value is only an estimate of the
  parameter mean value
• Parameter value is not known
• Due to sampling variability, no two samples will
  produce exactly the same outcome, or sample mean
•   
•       Can we estimate how this sample mean
  value would vary if you take many large samples
  from the same population?
•  
• Remember
•  
•       sample mean values from large samples
  have a normal distribution
•  
•       the mean of the sampling distribution is
  the same as the unknown parameter ?
•  
• standard deviation of for a SRS of size n is ?
  

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PARAMETRIC STATISTICAL INFERENCE ESTIMATION

• Example A random sample of 350 male college
  students were asked for the number of units they
  were taking. The mean was 12.3 units, with a
  standard deviation of 2.50 units.
•  
• What can we say about the mean number of units of
  all student males at the university? How will
  the estimate value of the parameter vary from one
  sample to another with a certain confidence, like
  95?
•  
• Assume that ? ?. s ?
•  
•  

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PARAMETRIC STATISTICAL INFERENCE ESTIMATION

• Statistical confidence
•  
• Remember The 68-95-99.7 rule
•  
•     In  95 of all samples, the mean score of x
  will lie within 2 standard deviations of the
  population mean score ?.
•  
• Since s 2.50, we can say that
•  
• In 95 of samples, ? will lie within 5.0 points
  of the observed sample mean
•  
• In 95 of all samples,
•  
•  
• Thus, the parameter will lie between 7.3 and
  17.3, in 95 of samples

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PARAMETRIC STATISTICAL INFERENCE ESTIMATION

• Rephrasing
•  
• 1.  We are 95 confident that the interval
  7.3-17.3 contains ?
•  
• We have just assigned statistical confidence to
  our estimation of the parameter
• We call this estimated interval a CONFIDENCE
  INTERVAL for the mean value

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PARAMETRIC STATISTICAL INFERENCE ESTIMATION

•    But, there is still some chance that the true
  parameter value will not lie in the identified
  interval
•  e.g. The SRS chosen was one of few samples for
  which is not within 5.0 points of true mean.
  5 of samples will give these incorrect results

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PARAMETRIC STATISTICAL INFERENCE ESTIMATION

•     CONFIDENCE INTERVAL formal definition
•  
• A level C confidence interval for a
  parameter is defined as
•  
• estimate ? margin of error
•  
• and gives the interval that will capture the
  true parameter value in repeated samples with a
  certain probability
•       Confidence intervals usually vary between
  90 and 99.9

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PARAMETRIC STATISTICAL INFERENCE ESTIMATION

• BUILDING CONFIDENCE INTERVALS
• If we know the parameter ? and ?, we can
  standardize the sample mean. The result is the
  ONE-SAMPLE Z STATISTIC
• The z statistic tells us how far the
  observed is from ?, in units of standard
  deviations of . Because has a normal
  distribution, z has the standard normal
  distribution N(0,1).

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PARAMETRIC STATISTICAL INFERENCE ESTIMATION

•     Constructing confidence intervals
• When we construct a 95 confidence interval, we
  are looking for two values for which there is a
  95 chance that the population mean is between
  them. So,
• P(Low lt ? lt High) 0.95
• Thus, 0.95 P(-1.96 lt z lt 1.96)
• 0.95

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PARAMETRIC STATISTICAL INFERENCE ESTIMATION

•     Draw a SRS of size n from a population
  having unknown mean ?, and known standard
  deviation ?. A level C confidence interval for ?
• This interval is exact when the population
  distribution is normal and is approximately
  correct for large n in other cases
• where ? represents the probability that the
  interval will not capture the true parameter
  value in repeated sample or confidence level, and
  C is the confidence level.

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Confidence intervals and confidence levels of
Standardized normal curve N(0,1)
Figure 6.5 and figure 6.6
z z?/2
C chosen confidence level probability that a
parameter will lie within a given interval with
a desired confidence (1-C)/2 probability that
a parameter will be situated either above or
below the the lower confidence limit ?/2
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PARAMETRIC STATISTICAL INFERENCE ESTIMATION

•   Example
• A manufacturer of pharmaceutical products
  analyzes a specimen from each batch of a product
  to verify the concentration of the active
  ingredient. The chemical analysis is not
  perfectly precise. Repeated measurements on the
  same specimen give slightly different results.
  The results of repeated measurements follow a
  normal distribution. The analysis procedure has
  no bias, so the mean of the population of all
  measurements is the true concentration in the
  specimen. The standard deviation of this
  distribution is known to be 0.0068 g/l. Three
  analyses of one specimen give the following
  concentrations 
• 0.8403 0.8363 0.8447
• Calculate the 99 confidence interval for the
  true concentration.

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PARAMETRIC STATISTICAL INFERENCE ESTIMATION

•      INTERVAL ESTIMATION OF ? WITH ? UNKNOWN
• ? replaced with estimate s introduces more
• uncertainty
• STUDENTS T-DISTRIBUTION
• not standard normal curve

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PARAMETRIC STATISTICAL INFERENCE ESTIMATION

• INTERVAL ESTIMATION OF ? WITH ? UNKNOWN
• Intervals derived from t-distribution are
  wider than those found with z-distribution
• For large samples (ngt30), it makes no
  difference which distribution we use to estimate
  confidence interval

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PARAMETRIC STATISTICAL INFERENCE ESTIMATION

• HOW CONFIDENCE INTERVALS BEHAVE
•  
•     Ideal situation high confidence and small
  margin of error
•  
• Margin of error (E)
•  
•     The smaller the margin of error, the more
  precise our estimation of ?

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PARAMETRIC STATISTICAL INFERENCE ESTIMATION

•    Properties of error
•  
• 1.  Error increases with smaller sample size
• For any confidence level, large samples
  reduce the margin of error
•  
• 2.  Error increases with larger standard
  Deviation
•      As variation among the individuals in the
  population increases, so does the error of our
  estimate
•  
• 3.   Error increases with larger z values
• Tradeoff between confidence level and margin
  of error
•   

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Interval width (error) increases with Increased
confidence level Higher confidence levels
have Higher z values
Figure 8-10 and 8-11
Error is high in small samples
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PARAMETRIC STATISTICAL INFERENCE ESTIMATION

• Example
•  
• Calculate the 99 confidence interval for sample
  size of 1. ? 0.8404, ? 0.0068
•   
• 99 confidence interval for n3 was 0.8303 to
  0.8505 g/l
•  
• How do these compare in relation to the mean?
  Which one has the larger margin of error?

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CHOOSING SAMPLE SIZE

•      Sometimes we wish to estimate our mean
  within a certain margin of error.
• Sometimes we wish to determine a certain sample
  size in order to achieve a given margin of error
• Here is how
•  
• Remember
•  
• Margin of error (E)
•  
• To obtain a desired value of E, for a given
• confidence level, you need to figure out n.
•  
• From the above,
•  
•  
•     It is the sample size that determines the
  margin of error
• Required sample size depends on the desired level
  of confidence

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CHOOSING SAMPLE SIZE
Example   Management asks the pharmaceutical
laboratory to produce results accurate to within
?0.005 with 95 confidence. How many
measurements must be averaged to comply with this
request?     m 0.005 g/l For 95 confidence
level, z ? ? 0.0068 g/l    
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CHOOSING SAMPLE SIZE
Example   Management asks the pharmaceutical
laboratory to produce results accurate to within
?0.005 with 95 confidence. How many
measurements must be averaged to comply with this
request?     m 0.005 g/l For 95 confidence
level, z 1.960. ? 0.0068 g/l     is n 7 or
n 8?   Choose one that will give a smaller
margin of error.   How should we always round to
meet the requirements necessary?
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SUMMARY

• All formulas for inference are only correct
  under certain conditions
• o    Most inference methods have several
  assumptions attached to them that must be met if
  the outcomes produced by them are to be reliable.
•  
• Confidence interval formula has the following
  assumptions
•  
• 1.   The data must come from a simple random
  sample.
• different methods exist for stratified and
  multistage samples
• undercoverage and non-response can add error
• 2. X bar must be a random normal variable
• 3.   There must be no outliers. Is the formula
  sensitive to outliers?
• 4.   If sample size is small (lt15) and/or ? is
  not known but distribution of x still normal,
  t-distribution must be used to compute interval
• 5.  When sigma is known use z-distribution.
• For large sample sizes we can assume that ?
  s and use either z or t distributions