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.
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.
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.
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.
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:
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
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 < .05 ).
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:
One-Tailed Hypothesis
If the scientific hypothesis predicts a direction of the results, we say it is a OneTailed 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: