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

valuesMost 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 α 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• α =.01 1% probability level• α =.05 5% probability level• α =.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 α 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 sample’s 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 isa significant difference between 2 ormore means at a specified significanceLevel (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 – α)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