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Inference and Inferential Statistics

- Methods of Educational Research
- EDU 660

Inference

- Draw conclusions from the data
- Allow researchers to generalize to a population

of individuals based on information obtained from

a sample of those individuals - Assesses whether the results obtained from a

sample are the same as those that would have been

calculated for the entire population

Probabilistic nature of inference

- How likely is it?
- Are the results that we have seen due to chance

or some real difference? - Mean score for 2 different groups
- Example
- X1 23.5 X2 31.6
- Is this a real difference between these
- scores?

Normal distribution

- A bell shaped curve reflecting the distribution
- of many variables of interest to educators

Normal distribution

- Characteristics
- 50 of the scores fall above the mean and 50

fall below the mean - The mean, median, and mode are the same values
- Most participants score near the mean the

further a score is from the mean the fewer the

number of participants who attained that score - Specific numbers or percentages of scores fall

between - ?1 SD, 68
- ?2 SD, 95
- ?3 SD, 99

Null and Alternative hypotheses

- The null hypothesis represents a statistical tool

important to inferential tests of significance - The alternative hypothesis usually represents the

research hypothesis related to the study

Null and Alternative hypotheses

- Comparisons between groups
- Null no difference between the means scores of

the groups - Alternative there are differences between the

mean scores of the groups - Relationships between variables
- Null no relationship exists between the

variables being studied - Alternative a relationship exists between the

variables being studied

Test of Significance

- Statistical analyses to help decide whether to

accept or reject the null hypothesis - Alpha a level
- An established probability or significance level

which serves as the criterion to determine

whether to accept or reject the null hypothesis - Common levels in education
- a .01 1 probability level
- a .05 5 probability level
- a .10 10 probability level

Type I and Type II Errors

- Correct decisions
- The null hypothesis is true and it is accepted
- The null hypothesis is false and it is rejected
- Incorrect decisions
- Type I error - the null hypothesis is true and it

is rejected - Type II error the null hypothesis is false and

it is accepted

Type I and Type II Errors

As a becomes smaller there is a smaller chance of

a Type 1 error but a greater chance of a Type 2

error.

One-Tailed and Two-Tailed Tests

- One-tailed an anticipated outcome in a specific

direction - Treatment group mean is significantly

higher/lower than the control group mean - Two-tailed anticipated outcome not directional
- Treatment and control groups are equal
- Ample justification needed for using one-tailed

tests

One-Tailed and Two-Tailed Tests

Test of Significance

- Specific tests are used in specific situations

based on the number of samples and the

statistics of interest - One sample tests of the mean, variance,

proportions, correlations, etc. - Two sample tests of means, variances,

proportions, correlations, etc.

Test of Significance

- Types of inferential statistics
- Parametric tests more powerful tests that

require certain assumptions to be met - t - tests
- ANOVA
- Non-parametric tests less powerful
- Chi-Square

Form a Null Hypothesis

- H0 There is no significant difference in the

mean scores for the 2 groups - Acceptance of the null hypothesis
- The difference between groups is too small to

attribute it to anything but chance - Rejection of the null hypothesis
- The difference between groups is so large it can

be attributed to something other than chance

(e.g., experimental treatment)

The t Test

- Used to test whether 2 means are significantly

different at a selected probability - The t test determines whether the observed

difference is sufficiently larger than a

difference that would be expected by chance

Types of t Tests

- t test for independent samples
- The members of one sample are not related to

those of the other sample in any systematic way -

come from the same population - Examples
- 1. Examine the difference between the mean

scores for an experimental and control group - 2. Examine the mean scores for men and women in

sample

Types of t Tests

- t test for NonIndependent samples
- Used to compare groups that are formed to examine

a samples performance on a single measure or

multiple measures - Example examining the difference between

pre-test and post-test mean scores for a single

class of students

Analysis of Variance - ANOVA

- ANOVA is used to test whether there is
- a significant difference between 2 or
- more means at a specified significance
- Level (usually 5)
- Example Is there a significant difference in

the mean scores on a test (µ1, µ2, µ3) of 3

classes of college students?

ANOVA

- Omnibus Null Hypothesis
- H0 µ1 µ2 µ3
- Note repeated use of numerous t tests for more

than 2 means will result in an increased

probability of type I errors - p 1 - (1 a)c where c is the number of t

tests

Analysis of Variance - ANOVA

- If an ANOVA determines that there is a

significant difference among a group of means,

what then? - Multiple comparison methods are used to determine

what means are different Scheffe test

Steps in Statistical Testing

- State the null and alternative hypotheses
- Set alpha level - 0.05, 0.01 etc
- Identify the appropriate test of significance
- Identify the test statistic
- Compute the test statistic and probability level
- Is the probability level less than the specified

probability? - Accept or reject hypothesis