Analyzing Statistical Inferences

How to Not Know Null

Agenda

• Inferential stats– Descriptive vs. Inferential– Ramifications of hypothesis testing– Tests of significance

• Action Research– Dissect Sanna paper– Outline paper– Discuss presentations

Teacher Salary Example

• Descriptive Stats– Range of salary distributions– Mean– Percentages of teachers with different levels of

experience and degrees

Teacher Salary Example

• What kinds of questions would we ask if we wanted to compare these 2 groups of teachers?

• Inferential Statistics

Understand the need for using inferential statistics to estimate likely

conclusions

• What Are Inferential Statistics?– Inferential statistics refer to certain

procedures that allow researchers to make inferences about a population based on data obtained from a sample.

– The term “probability,” as used in research, refers to the predicted relative frequency with which a given event will occur (e.g., p-value).

Descriptive vs. Inferential Stats

• Describe the data. • Means, variances,

frequencies.• Important precursor

to inferential stats

• Infer from a sample what is true of a population.

• Rely on descriptive stats.

• Ultimate goal - to draw accurate conclusions about the population

Descriptive Inferential

The Notorious Null

• What is it??– A statement about a relationship– No differences between groups– No relationships between variables

• What assumption should you always make about the null?

Assume the null is accurate

• Null hypothesis differs in most instances from the research hypothesis– which states that one method is expected

to be more effective than another

• Rejecting the null hypothesis provides evidence (but not proof) that the intervention had an effect

The Notorious Null

Hypothesis Testing

• Tests of significance ask this question:

Could these observations really have occurred by chance?

• Example of Jury Selectionp < .0000000000000000014

Probability

• Level of significance or p = probability of being wrong to reject the null (to state there is a true difference, but in reality the difference is from chance)

• In general, research should be p < .05 to be considered significant.

• The decision a researcher must make is:– whether to accept the null hypothesis or to

reject it

• There are four possibilities…

Decisions concerning rejecting the null hypothesis…

The true status of the null hypothesis…

True False

The

res

earc

her’s

dec

isio

n ab

out

the

null

hypo

thes

is…

True

False

CorrectType II

Error (β)

Type I Error (α)

Correct

Consider an example from a kitchen…

• You probably have a smoke alarm where you live.

• You have probably made microwave popcorn or toast that set off your alarm though you had no fire.

TYPE I error

• If you ever took the batteries out of your smoke alarm because you got so annoyed,…

• You ran the risk of having a fire but no alarm.

TYPE II error

• If you had a fire, but your alarm worked, you’re ok.

• If you had no fire, and no alarm, you’re also ok.

• Hence…

Put in cooking terms…

The true status of the kitchen…

No Fire Fire

The

sta

tus

of t

he s

mok

e al

arm

…

NoAlarm

Alarm

CorrectType II

Error (β)

Type I Error (α)

Correct

Back to the null hypothesis…

The true status of the null hypothesis…

True False

The

res

earc

her’s

dec

isio

n ab

out

the

null

hypo

thes

is…

True

False

CorrectType II

Error (β)

Type I Error (α)

Correct

Your Turn for Statistical Fun

• Create a null hypothesis regarding the effectiveness of 2 methods of instruction on student achievement.

• Using the chart on the previous slide, state what is occurring with this particular hypothesis.

• What are ramifications of incorrect decisions?

Probability

• Level of significance or p = probability of being wrong to reject the null (to state there is a true difference, but in reality the difference is from chance)

• In general, research should be p < .05 to be considered significant.

Things that effect p

• Difference between 2 groups

• Sampling and/or measurement error

• Size of the sample

Steps in using inferential statistics

1. Select the test of significance

2. Determine whether significance test will be two-tailed or one tailed

3. Select α (alpha), the probability level (usually <.05)

4. Compute the test of significance5. Consult table to determine the

significance of the results

How to determine p

Tests of Significance

Tests of Significance

• Statistical formulas that enable the researcher to determine if there was a real difference between the sample means

• Examples– t test– ANOVA– Chi-square

t test

• Used to determine whether two means are significantly different at a selected probability level

• Adjusts for the fact that the distribution of scores for small samples becomes increasingly different from the normal distribution as sample sizes become increasingly smaller

• Sample t-table

t test

• If the t value is equal to or greater than the table value, then the null hypothesis is rejected because the difference is greater than would be expected due to chance

Reminder…

• Don’t forget the purpose. You are going through this statistical rigmarole because you want to know

Could these observations really have occurred by chance?

ANOVA

• A comparison of the means for two or more groups

• Example - Do the mean scores differ for the groups using co-operative group, lecture, or web-based instruction?

• The assumption is that randomly formed groups of participants are chosen and are essentially the same at the beginning of a study on a measure of the dependent variable

ANOVA

• F value of ANOVA is similar to t-value in t-test.

• If F value is significant, you know there is a difference somewhere, but have to do post hoc tests to figure out where.

Chi-Square

• Tests differences in frequencies across different categories– Do mothers and fathers differ in their

support of a year-round school calendar?– Do the percentages of undergraduate,

graduate, and doctoral students differ in terms of their support for the new class attendance policy?

Some words about significancesignificance

• “Statistical significance” is a term that refers to some statistical criterion, usually the numerical value of some formula or calculation.

• “Practical significance” means its utility, and that is in the eyes of the beholder. What may be impractical to you or me may be very practical to someone else.

Practical Significance

• An Example – A new reading program shows improved comprehension scores that are statistically significant. However, it takes many hours and dollars to train teachers to use the program. Does it warrant buying the new program?

• Big Question: Is it practical to use the results?

“There is no magical or purely technical way to decide whether or not a statistically significant difference means you should do something different in your school. There are only tools that assist your judgment. There is no escape from using judgment.”

--Gerald W. BraceyReading Educational Research: How

To Avoid Getting Statistically Snookered

Practical Significance

Large sample sizes can produce a statistically significant result even though there is limited or no practical importance associated with the finding.

Effect Size

• Take into account variance, not just the means.

• Refers to the magnitude of a difference.

• Levels you should knowd ≥ .75 = large effect

d ~ .5 = moderate effect

d ~ .3 = small effect

• Good website on Effect Size http://www.cemcentre.org/renderpage.asp?linkID=30325016

Evaluation Criteria

• Basic descriptive statistics are needed to evaluate the inferential results

• Inferential analyses report statistical significance, not practical significance

• Inferential analyses do not indicate internal or external validity

• The results depend on sample sizes

Evaluation Criteria

• The appropriate statistical procedures are used

• The level of significance is interpreted correctly