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Chapter 12 A Primer for Inferential Statistics What Does Statistically 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: • 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: • Suppose we measure the difference in selfesteem between 12 year old males and females and get a statistically significant difference, with males having higher selfesteem • This means that the difference probably reflects a “true” difference in the selfesteem levels. Wrong: < 5% of the time. 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 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 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 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 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 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 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 Observed • 30 • 160 • 10 Frequency Expected 15.6 (200 x .078) 137.8 (200 x .689) 46.6 (200 x .233) • Degrees of Freedom = 2 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. Decision • 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-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.