Chapter 12 A Primer for Inferential Statistics What Does Statistically Significant Mean? It’s the probability that an observed difference or association.

Chapter 12Chapter 12

A Primer for Inferential Statistics

What Does Statistically What Does Statistically Significant Mean?Significant Mean?

• It’s the probability that an observed difference or association is a result of sampling fluctuations, and not reflective of a “true” difference in the population from which the sample was selected

Example 1:Example 1:

• Suppose we test differences between high school men and women in the hours they study: females spend 12 minutes more per night than males and the result is analyzed and shown to be statistically significant

• It means that less than 5% of the time could the difference be due to chance sampling factors

Example 2:Example 2:

• Suppose we measure the difference in self-esteem between 12 year old males and females and get a statistically significant difference, with males having higher self-esteem

• This means that the difference probably reflects a “true” difference in the self-esteem levels. Wrong: < 5% of the time.

Example 3:Example 3:

• You test the relation between gender and self-esteem: a test of significance indicates that the null hypothesis should be accepted. What does this mean?

• It means that more than 5% of the time the difference you are getting could be the result of sample fluctuations

Clinically Significance

• Clinical significance means the findings must have meaning for patient care in the presence or absence of statistical significance

• Statistical significance indicates that the findings are unlikely to result from chance, clinical significance requires the nurse to interpret the findings in terms of their value to nursing

Sample FluctuationSample Fluctuation

• Sample fluctuation is the idea that each time we select a sample we will get somewhat different results

• If we selected repeated samples, and plotted the means, they would be normally distributed; but each one would be different

A Test of Significance

• A test of significance reports the probability that an observed difference is the result of sampling fluctuations and not reflective of a “real” difference in the population from which the sample has been taken

Research & Null HypothesisResearch & Null Hypothesis

• Research Hypothesis: reference is to your predicted outcome.

• Null Hypothesis: the prediction that there is no relation between the variables.

• It is the null hypothesis that is tested

Testing the Null HypothesisTesting the Null Hypothesis

• In a test, you either accept the null hypothesis or you reject it.

– To accept the null hypothesis is to conclude that there is no difference between the variables

– To reject the null is to conclude that there probably is a difference between the variables.

One- and Two-Tailed TestsOne- and Two-Tailed Tests

If you predict the direction of a relationship,

you do a one-tailed test; if you do not predict

the direction, you do a two-tailed test.

• Example: females are less approving of violence than are males (one-tailed)

• Example: there is a gender difference in the acceptance of violence (two-tailed)

Type I & II Errors

• TYPE 1. Reject a null hypothesis (that states no relationship between variables) when it should be accepted

• TYPE 2. Accept a null hypothesis when it should be rejected

• RAAR -Reject when you should accept: Accept when you should reject-the first 2 letters give you type 1, the second two letters, type 2

Chi-Square: Red & White BallsChi-Square: Red & White Balls

• The Chi-square (X2) involves a comparison of expected frequencies with observed frequencies. The formula is:

X2 = (fo - fe)2

fe

One Sample Chi-Square TestOne Sample Chi-Square Test

Suppose the following incomes:

INCOME STUDENT GENERAL

SAMPLE POPULATION

Over $100,000 30 15.0 7.8

$40,000 - $99,999 160 80.0 68.9

Under $40,000 10 5.0 23.3

TOTAL 200 100.0 100.0

The Computation

• Remember, Chi-squares compare expected frequencies (assuming the null hypothesis is correct) to the observed frequencies.

• To calculate the expected frequencies simply multiply the proportion in each category of the general population times the total no. of students (200).

• Why do you do this?

Why?

• If the student sample is drawn equally from all segments of society then they should have the same income distribution (this is assuming the null hypothesis is correct).

• So what are the expected frequencies in this case?

Expected Frequencies fe

Frequency Frequency

Observed Expected

• 30 15.6 (200 x .078)

• 160 137.8 (200 x .689)

• 10 46.6 (200 x .233)

• Degrees of Freedom = 2

Decision:Decision:

• Look up Chi square value in Appendix p. 399• 2 degrees of freedom• 1 tailed test (use column with value .10)• Critical Value is 4.61• Chi-Square calculated 45.61• Decision: (Calculated exceeds Critical) Reject

null hypothesis

Standard Chi-Square Test

• Drug use by Gender

• 3 categories of drug use (no experience, once or twice, three or more times)

• row marginal times column marginal divided by total N of cases yields expected frequencies

• degrees of freedom = (row - 1)(columns - 1) = 2.

DecisionDecision

• With 2 degrees of freedom, 2-tailed test, the Critical Value is 5.99

• Calculated Chi-Square is 5.689

• Does not equal or exceed the Critical Value

• So, your decision is what?

• Accept the null hypothesis

T-TestsT-Tests

• Sample sizes < 30

• Dependent variable measured at ratio level

• Independent assignment to treatments

• Treatment has two levels only

• Population normally distributed

Two T-Tests: Between & Within

• Between-Subjects T-Test: used in an experimental design, with an experimental and a control group, where the groups have been independently established.

• Within-Subjects: In these designs the same person is subjected to different treatments and a comparison is made between the two treatments.