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## Inferential Statistics

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### Inferential Statistics Statistical Analysis of Research Data * * * * * * * * * * * * * * * * * * * * * * * * * Statistical Inference Getting information about a ... – PowerPoint PPT presentation

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Title: Inferential Statistics

1
Inferential Statistics
• Statistical Analysis of Research Data

2
Statistical Inference
• Getting information about a population from a
sample
• How practical are statistically significant
results?
• Cost/benefit
• Crucial difference
• Client acceptability
• Public and political acceptability
• Ethical and legal concerns

3
Inferential Statistics
• These provide a means for drawing conclusions
about a population given the data actually
obtained from the sample. They allow a
researcher to make generalizations to a larger
number of individuals, based on information from
a limited number of subjects. They are based on
• Probability theory
• Statistical inference
• Sampling distributions

4
Inferential Statistics
• Probability theory the basis for
decision-making statistical inferences. It refers
to a large number of experiences, events or
outcomes that will happen in a population in the
long run. Likelihood and chance are similar
terms. Examples are usually based on tossing a
coin and finding heads or tails. Probabilities
are statements of likelihood expressed in values
from 0 to 1.0.
• p the number of outcomes
• the total possible outcomes

5
Inferential Statistics
• Statistical inference statistics enable us to
judge the probability that our inferences or
estimates are close to the truth
• Sampling distributions are theoretical
distributions developed by mathematicians to
organize statistical outcomes from various sample
sizes so that we can determine the probability of
something happening by chance in the population
from which the sample was drawn. They allow us
to know the relative frequency or probability of
occurrence of any value in the distribution.

6
Inferential Statistics
• Hypothesis testing 5 basic steps
• Make a prediction
• Decide on a statistical test to use
• Select a significance level and a critical region
(region of rejection of the null hypothesis). To
do this you must consider two things
• Whether both ends (tails) of the distribution
should be included.
• How the critical region of a certain size will
contribute to Type I or Type II errors.

7
Critical Region in the Sampling Distribution for
a One-Tailed Test
8
Critical Regions in the Sampling Distribution for
a Two-Tailed Test
9
Levels of Significance
• Remember, if a printsout shows a two-tailed test
result, and you wanted a one-tailed result,
divide the two tailed p value by 2.
• Example p .080 (two-tailed) or pgt.05
• p .040 (one-tailed) or plt.05
• The first would not be statistically significant,
whereas, the second would be statistically
significant

10
Outcomes of Statistical Decision Making
11
Inferential Statistics
• Hypothesis testing cont.
• Computing the test statistic - The test statistic
is not a mean, sd or any form of descriptive
data. It is simply a number that can be compared
with a set of results predicted by the sampling
distribution
• Compare the test statistic to the sampling
distribution (table) and make a decision about
the null hypothesis reject it if the statistic
falls in the region of rejection.
• Consider the power of the test its probability
of detecting a significant difference
parametric tests are more powerful

12
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13
Degrees of Freedom
• One sample t-test or paired t-test N-1
• Independent t-test N-2
• Chi-square test
• ( rows - 1) x ( columns 1)
• ANOVA
• df between groups ( levels or groups
1)
• df within groups ( subjects - of
levels)
• Correlations N-2

14
Levels of Measurement
• There are four levels or scales of measurement,
Each level is classified according to certain
characteristics. Data that fall in the first
level are limited to certain statistical tests.
Choices of statistical tests (and the power of
the tests) increase as the levels go up.

15
Levels of Measurement cont.
• Nominal scale measurement at its weakest
numbers or other symbols are used to classify or
partition a class into mutually exclusive
subclasses animals can be classified as dogs,
cats, etc. You can test hypotheses regarding
distribution among the categories by using the
Chi-square test.

16
Levels of Measurement cont.
• Ordinal scale shows relationships among
classes, such as higher than, more difficult
than, etc. It allows the attributes of a variable
to be ranked in relation to each other. A
researcher can test hypotheses using
non-parametric statistics of order and ranking.

17
Levels of Measurement cont.
• Interval scale is similar to the ordinal scale,
but the distance between any two numbers is of a
known size. The numbers used have absolute values
and the interval between each number is
considered to be equal. Increasing amounts of a
variable are represented by increasing numbers on
the scale. The variable is continuous. There is
no true zero where you have none of the
variable. All parametric tests can be used with
interval data.

18
Levels of Measurement cont.
• Ratio scale is like the interval scale but it
has a true zero point as its origin. Time,
length and weight are ratio scales when used
alone, but not as a characteristic of a person.
Arithmetic, all parametric tests and geometric
means can be used with ratio data.

19
Tests of Significance
• Parametric tests of significance used if there
are at least 30 observations, the population can
be assumed to be normally distributed, variables
are at least in an interval scale
• Z tests are used with samples over 30. There are
four kinds (two samples or two categories)
• t-tests are used when samples are 30 or less.
• Single sample t-test (one sample)
• Independent t-test (two samples)
• Paired t-test (two categories

20
Tests of Significance
• Non-parametric tests of significance small
numbers, cant assume a normal distribution, or
measurement not interval
• Chi-square requires only nominal data allows
researcher to determine whether frequencies that
have been obtained in research differ from those
that would have been expected use a X2 sampling
distribution
• Chi-square goodness of Fit
• Chi-Square test of independence

21
Tests of Significance
• Mann Whitney U an alternate to the independent
t-test must have at least ordinal data. It
counts the comparative ranks of scores in two
samples (from highest to lowest) The null
hypothesis is that the two samples are randomly
distributed. Use U sampling distribution tables
for small sample sizes (1-8) and medium sample
sizes (9-20) and the Z test for large samples

22
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23
Tests of Significance
• Wilcoxin Matched Pairs (signed rank test) is an
alternate to the paired t-test. It is used for
repeated measures on the same individual. It
requires a measurement between ordinal and
interval scales the scores must have some real
meaning. Use a T table. If the T is less than or
equal to the T in the table, the null hypothesis
is rejected.

24
Measures of Association
• Parametric Measures of Association These answer
the question, within a given population, is
there a relationship between one variable and
another variable? A measure of association can
exist only if data can be logically paired. It
can be tested for significance.
• Correlation answers the question, What is the
degree of relationship between x and y Use
Pearson Product Moment Correlation (Pearson r )
see next slide

25
Measures of Association The Pearson Correlation
Coefficient (Pearson r)
• The r examines the relationship between two
quantitative sets of scores.
• The r varies from 1.00 to 1.00
• The r is not a proportion and cannot be
multiplied by 100 to get a percentage.
• To think of the r as a percentage, it needs to be
converted to the Coefficient of Determination
or R2 . An r of .50 is 25 better than an r of
0.00

26
Measures of Association
• Non-parametric tests for association
• Correlation
• The Spearman Rank Order Correlation (Rs) To
what extent and how strongly are two variables
related?
• Phi coefficient it can be used with nominal
data, but should have ordinal data
• Kendalls Q can be used with nominal data

27
Prediction
• Parametric Prediction using a correlation, if
you know score x, you can predict score y for
one person Use regression analysis
• Simple linear regression allows the prediction
from one variable to another you must have at
least interval level data
• Multiple linear regression this allows the
prediction of one variable from several other
variables. The dependent variable must be on the
interval scale

28
Prediction
• Non-parametric Prediction measures the extent
to which you can reduce the error in predicting
the dependent variable as a consequence of having
some knowledge of the independent variable such
as, predicting income DV by education IV
• Kendalls Tau used with ordinal data and
ranking - is better than the Gamma because it
takes ties into account
• Gamma - used with ordinal data to predict the
rank of one variable by knowing rank on another
variable
• Lambda can be used with nominal data
knowledge of the IV allows one to make a better
prediction of the DV than if you had no knowledge
at all

29
Parametric Multiple Comparisons
• The analysis of variance (ANOVA) is probably the
most commonly encountered multiple comparison
test. It compares observed values with expected
values in trying to discover whether the means of
several populations are equal. It compares two
estimates of the population variance. One
estimate is based on variance within each sample
within groups. The other is based on variation
across samples between groups. The between
group variance is the explained variance (due to
the treatment) and the variation within each
group is the unexplained variance (the error
variance).

30
Parametric Multiple Comparisons
• ANOVA cont. The ratio of the explained scores to
the unexplained scores gives the F statistic. If
the variance between the groups is larger, giving
an F ratio greater than 1, it may be significant
depending upon the degrees of freedom. If the F
ratio is approximately 1, it means that the null
hypothesis is supported and there was no
significant difference between the groups.

31
Parametric Multiple Comparisons
• ANOVA cont. If the null hypothesis is rejected,
then one would be interested in determining which
groups showed a significant difference. The best
way to check this is to conduct a post hoc test
such as the Tukey, Bonferrioni, or Scheffe. (SPSS
will do this for you if you click on Post-hoc and
check the test desired.Check on descriptives
while you still in ANOVA, and the program will
also give you the mean for each group)

32
Parametric Multiple Comparisons
• Two-Way Analysis of Variance
• Classifies participants in two-ways
• Two main effects
• An interaction effect

33
Non-parametric Multiple Comparison
• Kruskal-Wallis Test an alternative to the
one-way ANOVA. The scores are ranked and the
analyses compare the mean rank in each group. It
determines if there is a difference between
groups.
• McNemar Test an adaptation of the Chi-square
that is used with repeated measures at the
nominal level.
• Friedman Test an alternative to the repeated
ANOVA. Two or more measurements are taken from
the same subjects. It answers the questions as to
whether the measurement changes over time.