A Primer for Inferential Statistics

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Chapter 12

• A Primer for Inferential Statistics

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What Does Statistically Significant Mean?

• Its 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

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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

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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 lt 5 of the time.

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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

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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

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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

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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

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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

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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.

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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)

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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

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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
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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

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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?

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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?

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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

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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

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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.

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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

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T-Tests

• Sample sizes lt 30
• Dependent variable measured at ratio level
• Independent assignment to treatments
• Treatment has two levels only
• Population normally distributed

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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.