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Inferential Statistics

- Statistical Analysis of Research Data

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

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

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

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.

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.

Critical Region in the Sampling Distribution for

a One-Tailed Test

Critical Regions in the Sampling Distribution for

a Two-Tailed Test

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

Outcomes of Statistical Decision Making

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

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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

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.

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.

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.

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.

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.

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

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

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

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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.

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

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

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

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

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

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).

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.

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)

Parametric Multiple Comparisons

- Two-Way Analysis of Variance
- Classifies participants in two-ways
- Results answer three questions
- Two main effects
- An interaction effect

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.