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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