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Introduction to Inferential Statistics

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Title: Introduction to Inferential Statistics


1
Introduction to Inferential Statistics
  • Statistical analyses are initially divided into
  • Descriptive Statistics or Inferential
    Statistics.
  • Descriptive Statistics are used to organize
    and/or summarize the parameters associated with
    data collection (e.g., mean, median, mode,
    variance, standard deviation)
  • Inferential Statistics are used to infer
    information about the relationship between
    multiple samples or between a sample and a
    population (e.g., t-test, ANOVA, Chi Square).
  • The Scientific Method uses Inferential Statistics
    to determine if the independent variable has
    caused a significant change in the dependent
    variable and if that change can be generalizable
    to the larger population.
  • If a dependent variable within the population is
    normally distributed and we can calculate or
    estimate the mean and standard deviation of the
    population, then we can use probabilities to
    determine significance of these relationships. 
  • To begin we must collect a representative sample
    of our much larger population.
  •  

2
Sampling
  • A Representative Sample means that all
    significant subgroups of the population are
    represented in the sample. 
  • Random Sampling is used to increase the chances
    of obtaining a representative sample. It assures
    that everyone in the population of interest is
    equally likely to be chosen as a subject. 
  • Random Number Generators or Tables are used to
    select random samples.  Each number must have the
    same probability of occurring as any other
    number.  The larger the random sample, the more
    likely it will be representative of the
    population. 

3
Sampling Distribution
  • Generalization refers to the degree to which the
    mean of a representative sample is similar to or
    deviates from the mean of the larger population.
    This generalization is based on a distribution of
    sample means for all possible random samples.
  • This distribution is called a Sampling
    Distribution.
  • It describes the variability of sample means that
    could occur just by chance and thereby serves as
    a frame of reference for generalizing from a
    single sample mean to a population mean.
  • It allows us to determine whether, given the
    variability among all possible sample means, the
    one observed sample mean can be viewed as a
    common outcome (not statistically significant) or
    as a rare outcome (statistically significant).
  • All possible random samples of even a modest size
    population would consist of a very large number
    of possibilities and would be virtually
    impossible to calculate by hand. Therefore we use
    statistical theory to estimate the parameters.

4
Sampling Distribution
  • The Central Limit Theorem states that the
    distribution of sample means approaches a normal
    distribution when n is large. 
  • In such a distribution of unlimited number of
    sample means, the mean of the sample means will
    equal the population mean.

5
Standard Error of the Mean
  • The standard deviation of the distribution of
    sample means is called the Standard Error of the
    Mean or Standard Error for short. It is
    represented by the following formula
  •  
  • Since the standard deviation of the population is
    often unavailable, a good estimate of the
    standard error uses the standard deviation of the
    sample (s). This newly modified formula of the
    Standard Error is shown below
  •  

6
Statistical Significance
  • Now that we have the parameters of the Sampling
    Distribution we can see how to use this
    distribution to determine if the mean difference
    between two samples or between a sample and a
    population are significantly different from each
    other.
  • The Research Hypothesis (H1) states that the
    sample means of the groups are significantly
    different from one another. 
  • The Null Hypothesis (H0) states that there is no
    real difference between the sample means.
  •  
  • Anytime you observe a difference in behavior
    between groups, it may exist for two reasons
  • 1.) there is a real difference between the
    groups, or
  • 2.) there is no real difference between the
    groups, the results are due to error involved in
    sampling. 
  • This error can be described in two ways
  • Type I error is when you reject the null
    hypothesis when shouldn't have
  •  
  • Type II error is when you fail to reject the null
    hypothesis when you should have

7
Statistical Significance
  • The probability of committing a Type I error is
    designated by alpha.
  • An alpha level of 0.05 is reasonable and widely
    accepted by all scientists.  
  • The null hypothesis can be rejected if there is
    less than 0.05 probability of committing a Type I
    error ( p lt .05 ).

8
Two-Tailed Hypothesis
  • If the research hypothesis does not predict a
    direction of the results, we say it is a
    Two-Tailed Hypothesis because it is predicting
    that alpha will be split between both tails of
    the distribution. If the sample mean falls in
    either of these areas we can reject the null
    hypothesis. This is shown below

9
One-Tailed Hypothesis
  • If the scientific hypothesis predicts a direction
    of the results, we say it is a One-Tailed
    Hypothesis because it is predicting that alpha
    will fall only in one specific directional tail.
    If the sample mean falls in this area we can
    reject the null hypothesis. This is shown below
  •  
  •  
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