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## Introducing Inferential Statistics

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### Descriptive statistics provide basic measures of a distribution of scores ... A NEW TWIST TO p .05. A statement of probability, e.g., p .05 ... – PowerPoint PPT presentation

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Title: Introducing Inferential Statistics

1
Chapter 8
• Introducing Inferential Statistics

2006 Prentice Hall, Salkind.
2
CHAPTER OVERVIEW
• Say Hello to Inferential Statistics
• The Idea of Statistical Significance
• Tests of Significance
• Significance Versus Meaningfulness
• Meta-analysis

3
SAY HELLO TO INFERENTIAL STATISTICS
• Descriptive statistics provide basic measures of
a distribution of scores
• Inferential statistics allow inferences to a
larger population from the sample

4
HOW INFERENCE WORKS
• Representative samples from two groups are
selected
• Participants are tested
• Means from each group are compared
• Researcher concludes that measured differences
between groups either
• Result from chance, or
• Reflect true differences
• A conclusion is drawn regarding the role group
membership plays in observed differences

5
THE ROLE OF CHANCE
• Chance is the first explanation for observed
differences
• Chance is unexplained variability
• The goal of science is to
• Control sources of variability, thus
• Reducing the role of chance as an explanation

6
THE CENTRAL LIMIT THEOREM
• The means of samples drawn from a population will
be normally distributed
• This is so regardless of the shape of the
population distribution
• This demonstrates the power of inference

7
AN EXAMPLE OF THE CENTRAL LIMIT THEOREM
8
THE IDEA OF STATISTICAL SIGNIFICANCE
• Because sampling is imperfect
• Samples may not ideally match the population,
and
• Because hypotheses cannot be directly tested
• Inference is subject to error

9
STATISTICAL SIGNIFICANCE
• The degree of risk that you are willing to take
that you will reject a null hypothesis when it is
actually true

10
MAKING A DECISION
11
TYPE I AND TYPE II ERRORS
• The probability of making a Type I error
• Set by researcher
• e.g., .01 1 chance of rejecting null when it
is true
• e.g., .05 5 chance of rejecting null when it
is true
• Not the probability of making one or more Type I
errors on multiple tests of null!
• The probability of making a Type II error
• Not directly controlled by researcher
• Reduced by increasing sample size

12
HOW A TEST OF SIGNIFICANCE WORKS
• Each type of null hypothesis is tested with a
particular statistic
• Each statistic is characterized by a unique
distribution of values that are used to evaluate
the sample data

13
USING A STATISTICAL TEST
• State the null hypothesis
• Ho µ 1 µ2
• Establish significance level
• e.g., p .05
• e.g., p .01

14
USING A STATISTICAL TEST
• Select appropriate test statistic
• Compute test statistic (obtained value)
• Determine value needed to reject null (critical
value), which depends on
• Level of significance chosen (e.g., p 0.5)
• Degrees of freedom (based on sample size)
• Compare obtained value to critical value
• If obtained value gt critical value, reject null
• If obtained value ? critical value, accept null

15
t-TEST FOR INDEPENDENT MEANS Used to test null
hypothesis when two independent, unrelated groups
are compared E.g., Chen and Stevenson (1989)
• State null
• Establish level of risk
• Select test statistic
• Compute value
• Determine critical value
• Compare obtained value
• Ho µ 1980 µ 1984
• p .05
• t-test
• 2.00
• 1.980
• 2.00 gt 1.980 p lt .05

16
WHAT DOES t120 2.00, p lt .05 Really Mean?
• t type of test
• 120 degrees of freedom
• (related to sample size)
• 2.00 obtained value of t test
• p probability
• .05 level of significance
• (Type I error rate)

17
A NEW TWIST TO p lt .05
• A statement of probability, e.g.,
• p lt .05
• indicates that the probability of making a Type
I error on a test is less than .05
• But SPSS and other data analysis software compute
exact probabilities, e.g.,
• p .0375

18
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19
LOOKING AT DIFFERENCES BETWEEN GROUPS
20
t-TEST FOR DEPENDENT MEANS
• State null
• Establish level of risk
• Select test statistic
• Compute value
• Determine critical value
• Compare obtained value
• Ho µ 1A µ 1B
• p .01
• t-test
• 4. 2.581
• 5. 2.771
• 2.581 lt 2.771
• p gt .01

21
LOOKING AT RELATIONSHIPS BETWEEN VARIABLES
22
MULTIVARIATE ANALYSIS OF VARIANCE (MANOVA)
• Simultaneously tests differences between groups
on multiple dependent variables, but
• Because dependent variables might be related
• True Type I Error rate is inflated
• 1 (1 - )k
• Type I error rate
• k number of pairwise comparisons
• So, MANOVA takes these possible relationships
into account

23
FACTOR ANALYSIS
• A factor groups several related measures into one
construct
• The new construct is treated as a dependent
variable
• This technique allows a researcher to more
efficiently examine how these sets of variables
are related

24
SIGNIFICANCE VERSUS MEANINGFULNESS
• Statistical significance refers to the
• Probability that chance influenced observed
differences
• Not to the meaningfulness or importance of
observed differences
• Statistical significance must be interpreted
within a larger context

25
META-ANALYSIS
• Compares the results of multiple independent
studies that have examined the same conceptual,
dependent variable
• Allows examination of trends and patterns that
may exist in many different groups in many
different studies

26
HOW META-ANALYLSES ARE DONE
• An adequate sample of studies is collected
• Results from these studies are converted to a
common measureusually effect size
• Important aspects of the study are coded
• Descriptive and correlational techniques are used
to look for trends or common patterns in the
outcomes of the group of studies