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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.