Topics: Inferential Statistics

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Topics Inferential Statistics

• Inference
• Terminology
• Central Limit Theorem
• Estimation
• Point Estimation
• Confidence Intervals
• Hypothesis Testing

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Inferential Statistics

• Research is about trying to make valid inferences
• Inferential statistics the part of statistics
  that allows researchers to generalize their
  findings beyond data collected.
• Statistical inference a procedure for making
  inferences or generalizations about a larger
  population from a sample of that population

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How Statistical Inference Works
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Basic Terminology

• Population any collection of entities that have
  at least one characteristic in common
• Parameter the numbers that describe
  characteristics of scores in the population
  (mean, variance, s.d., correlation coefficient
  etc.)

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Basic Terminology (contd)

• Sample a part of the population
• Statistic the numbers that describe
  characteristics of scores in the sample (mean,
  variance, s.d., correlation coefficient,
  reliability coefficient, etc.)

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Basic Statistical Symbols
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Basic Terminology (cont)

• Estimate a number computed by using the data
  collected from a sample used as a best guess
  about a population parameter
• Estimator formula used to compute an estimate

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The Process of Estimation
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Types of Samples

• Probability
• Simple Random Samples
• Simple Stratified Samples
• Systematic Samples
• Cluster Samples
• Non Probability
• Purposive Samples
• Convenience Samples
• Quota Samples
• Snowball Samples

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Evaluating Samples

• If randomness of sample is impaired by refusal to
  participate
• Is participation rate reasonably high?
• Is there reason to believe participants and
  non-participants are similar on the relevant
  variables?
• If sample was not random
• Is it drawn from target group for the
  generalization?
• Is it at least reasonably diverse?
• Does researcher explicitly discuss this
  limitation?
• Has author described relevant demographics of
  sample?
• Is sample size sufficiently large?

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Limits on Inferences and Warnings

• Response Rates
• Source of data
• Sample size and sample quality
• Random

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Estimation

• Point Estimation
• Interval estimation
• Sampling Error
• Sampling Distribution
• Confidence Intervals

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Interval Estimation

• Interval Estimation an inferential statistical
  procedure used to estimate population parameters
  from sample data through the building of
  confidence intervals
• Confidence Intervals a range of values computed
  from sample data that has a known probability of
  capturing some population parameter of interest

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Key Concepts

• Sampling Distribution
• Sampling Error
• Standard Error
• Confidence Interval

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Central Limit Theorem

• The sampling distribution of means, for samples
  of 30 or more
• Is normally distributed (regardless of the shape
  of the population from which the samples were
  drawn)
• Has a mean equal to the population mean, mu
  regardless of the shape population or of the size
  of the sample
• Has a standard deviation--the standard error of
  the mean--equal to the population standard
  deviation divided by the square root of the
  sample size

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Sampling Distribution of Mean
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Central Limit Theorem Again

• The sampling distribution of means, for samples
  of 50 or more
• Is normally distributed (regardless of the shape
  of the population from which the samples were
  drawn)
• Has a mean equal to the population mean, mu
  regardless of the shape population or of the size
  of the sample
• Has a standard deviation--the standard error of
  the mean--equal to the population standard
  deviation divided by the square root of the
  sample size

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Sampling Distribution and Sampling Error Under CTL
u
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mu
Population mean
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Sampling Distribution and Standard Error

• Sampling Distribution a theoretical distribution
  that shows the frequency of occurrence of values
  of some statistic computed for all possible
  samples of size n drawn from some population.
• Sampling Distribution of the Mean A theoretical
  distribution of the frequency of occurrence of
  values of the mean computed for all possible
  samples of size n from a population
• Standard Error the standard deviation of the
  sampling distribution of the statistic

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Review Different Version of Standard Deviations

• Standard deviation
• Standard error of measurement
• Standard error of a statistic (mean, variance,
  correlation coefficient, t-statistic etc)

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Building Confidence Interval (95)
95
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98.5 97 98.5
100
101.5 103 104.5
99
105
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Confidence Intervals (CI)

• A range of values having a known probability that
  the interval computed from the sample data
  includes the population parameter of interest
• A defined interval of values that includes the
  statistic of interest, by adding and subtracting
  a specific amount (in this case standard error
  points) from the computed statistic (in this case
  the sample mean)

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Process for Constructing Confidence Intervals

• Compute the sample statistic (e.g. a mean)
• Compute the standard error of the statistic
  (mean)
• Make a decision about level of confidence that is
  desired (usually 95 or 99)
• Identify table value for 95 or 99 confidence
  interval
• Multiply standard error of the mean by the tabled
  value
• Form interval by adding and subtracting
  calculated value to and from the mean

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Various Levels of Confidence

• When population standard deviation is known use Z
  table values
• For 95CI mean /- 1.96 s.e. of mean
• For 99 CI mean /- 2.58 s.e. of mean

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Factors Affecting Width of Confidence Intervals

• Level of confidence
• Data variability
• Sample size

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Effects of Confidence Interval
95 times out of 100 the interval
constructed around the sample mean will
capture the population mean. 5 times out of 100
the interval will not capture the population mean
99 times out of 100 the interval
constructed around the sample mean will
capture the population mean. 1 time out of 100
the interval will not capture the population mean
99
95
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2.58sem
1.96sem
mu
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Effects of Variability
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Effects of Sample Size
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Practice Example What does the 95 CI really
mean?

• Sample of 1000 California seniors
• A sample mean on test of US History of 489
• Population (all California seniors) mean not
  known
• Known standard deviation of 100
• Pick the critical value for the Confidence
  Interval
• For 95 CI (with population s.d.known) /-
  1.96
• Calculate Standard Error of Mean
• SEMSD/Sqrt(N)100/Sqrt(1000)100/31.63.16
• Create a 95 Confidence Interval
• Adding and subtracting 1.96(3.16) 6.27 to 489
  creates a 95 confidence interval (CI) 482.73 -
  495.27