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Testing Hypotheses: Tails and Proportions

Distinction between NON-DIRECTIONAL and DIRECTIONAL research hypotheses

  • Non-directional hypotheses
    • Only state that one group differs from another on some characteristic, i.e., it does NOT specify the DIRECTION of the difference
    • Example: H0 -- Northwestern students differ from the college population in ideological attitudes
  • Directional hypotheses
    • Specifies the nature of the difference, i.e., that one group is higher, or lower, than another group on some attribute
    • Example:
      • NU students are more conservative than other students = H1
      • NU students are more liberal than other students = H2

DIRECTIONS and TAILS in hypotheses and statistical tests

Non-directional hypotheses use two-tailed tests

  • Any evidence of difference between NU students and the population supports a non-directional research hypothesis. 
    • The appropriate test is against the null hypothesis, H0: = 0 
    • Values different from 0, in either direction, are used in computing the test statistic, a z-value (or t, depending on sample size) 
  • Either large positive z-scores or large negative z-scores can lead to the rejection of the null hypothesis 
  • Thus the regions of rejection must lie in both tails of the normal distribution 
  • Assuming an alpha level of .05:
    • the rejection region to the right is marked by the critical value of +1.96 and contains .025 of the cases
    • that to the left is at -1.96 and also contains .025 of cases
  • Hence, a test of a non-directional hypothesis is a two-tailed test

Directional hypotheses use one-tailed tests

  • Directional research hypothesis: NU students are more conservative (higher on the scale) than other students-- H1: > 0
  • Any finding that shows NU students to be more liberal (lower on the scale) would directly contradict the research hypothesis
  • The proper test is against this "null" hypothesis-- H0 < (or =) 0
    • Now, only large positive z-scores can reject this null hypothesis.
    • Assuming an alpha level of .05:
      • the region of rejection is fixed entirely in the right- hand tail of the distribution
      • the right-hand tail alone must now contain .05 of the cases
      • the critical value now becomes a z-score of +1.65
        • .4505 cases lie between 0 and 1.65
        • .0495 (close enough to .05) lie to the right of 1.65
  • The size of the region of rejection remains the same (.05), but it lies only in one tail of the distribution, so it is marked by a smaller critical value: 1.65 < 1.96.
  • Hence, a one-tailed test offers a better chance to reject your null hypothesis

Errors in making statistical decisions

Type I error: The probability of rejecting a true null hypothesis is equal to the alpha level.
Type II error: The probability of accepting the null hypothesis when it is false (the beta value) is not easily calculated
As stated in the syllabus:

 Type I and Type II errors . . . are hard to keep straight, and even most researchers have to think hard before explaining the difference. An analogy with diagnosing anthrax may help.

Type I: A doctor rejects the hypothesis that the patient has anthrax and fails to prescribe cipro. The patient did have it and died.
Type II: A doctor accepts the hypothesis that the patient has anthrax and prescribes cipro. The patient didn't have it, cipro destroyed the patients' immune system, and the patient died of influenza.

Comments on directional hypotheses
  • It requires more theory to predict direction as well as difference
  • Because the predictions are more precise, you can count more outcomes against your hypothesis: i.e., those in the wrong direction
  • This permits using one-tailed tests, which do not need as much difference between observed and predicted values to yield significance
  • Moral: formulate directional hypotheses and use one-tailed tests when possible, e.g., in your research papers.