Inferential Statistics

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Inferential Statistics
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Sample and Population
real
Population PARAMETER
hypothetical
random/probability
Sample STATISTICS
non-random
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Why? and How?

• Why?
• To INFER from the sample data (STATISTIC) - what
  we have
• to the population (PARAMETER) -
  what we want
• To account for/handle the problem of SAMPLING
  ERROR
• To make decisions in the face of uncertainty, is
  something going on.

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How?

• Hypothesis Testing
• By determining the probability that the result
  obtained (based on the Descriptive Statistics)
  can be accounted for by ERROR (cause by the
  sampling method)
• via testing Statistical Hypotheses - using
  theoretical models (Sampling distributions) that
  represent what would be likely to happen, simply
  due to ERROR.

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How?

• Estimation
• By determining the precision of our ESTIMATE (we
  dont formally test an Hypothesis)
• via establishing CONFIDENCE INTERVALS around
  SAMPLE STATISTICS (where the range of the
  interval is primarily a function of the magnitude
  of the sampling error)
• What is the population value likely to be

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The concept of theoretical SAMPLING DISTRIBUTION

• A sampling distribution is a theoretical
  distribution of all possible
• sample values of a given size (n), under the
  assumption that the
• null-hypothesis is true (i.e.., under the
  assumption that the variability
• in the sample values is due to ERROR.
• The variability in the sample values is
  represented by the SE
• (Standard Error)

SAMPLING DISTRIBUTION Distribution of all
possible SAMPLE values of a given size
Mean
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How we use a SAMPLING DISTRIBUTION and STANDARD
ERROR (SE)

• We test the model represented by the sampling
  distribution by finding the relative position of
  our (one and only) real SAMPLE STATISTIC in
  this theoretical sampling distribution

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The Logic

• Common outcome
• If the model is true, then the relative position
  of our sample value in the model should not be
  far out (since, the further out, the less
  probable
• vs
• Rare outcomes
• If the relative position of our sample value is
  far out, (assuming it is, in fact a random
  representative sample from the population),
  then the model (which, recall, represents ERROR
  only) is probably not true.

Variability due to error

Relative position of our one, real Sample
Statistic
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Variability due to error

Relative position of our one, real Sample
Statistic
If this model were true, then is an improbable
event. Since is real, i.e.., since our sample
value is an empirical fact therefore, the
model is probably not true. So we reject the
model.
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Sampling distribution

• Inferential statistics estimates the population
  parameters from the sample values
• Characteristics
• A batch of means, called sampling distribution of
  means
• The mean of the sampling distribution equals the
  mean of the population
• The standard error of the mean is the standard
  deviation of the population
• The batch of sample means would be normally
  distributed around the mean of the distribution,
  X with a standard deviation of ????

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Central Limit Theorem

• if a population has a mean ??and a standard
  deviation ?? then the distribution of sample
  means drawn from this population approaches a
  normal distribution as N increases, with a
  and standard deviation ?????

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Significance Level

• When do we draw the line?
• At what point do we say that our result is a rare
  occurrence and that is highly unlikely to be
  sampling error?
• Significance level is the probability that a
  result is due to sampling error, and, if this
  probability is small enough, we reject the notion
  that sampling error is the cause.
• .05 significance level. If the probability that
  our result happened by change is .05 or less, we
  say that our results are significant at the .05
  level.

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Estimation

• The population mean is a fixed value, and it is
  the sample mean that deviate about this fixed
  value.
• Instead of talking of possible values that ??may
  take, given our sample X, we set up a confidence
  interval in which the true mean probably lies.
• 95 confidence interval - X 1.96sx
• X the sample mean and S standard error of the
  mean
• (sample mean 69.2 and s of 0.3)
• X 1.96sx 69.2 1.96(0.3)
• 69.2 .59
• 68.61 to 69.79

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Confidence interval -Examples

• Estimate of population mean
• N (200), X 102, S 12
• 95 confidence interval X 1.96sx
• First calculate Standard error of the mean
• X 1.96 102 1.96(0.85)
• 102 1.67
• 100.33 to 103.67

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• Claim that college women are taller than 10 years
  ago.
• Todays average height is 66
• We take a random sample of 50 college women
• We find a Mean 65, S 2.5
• Calculate SE
• We have to deny that the claim that average is 66
  is wrong, since the population mean is in the
  64.07 to 65.93 and 66 lies outside that interval.

X 2.58 65 2.58(0.36) 65
0.93 64.07 to 65.93