Two-Tailed vs. One-Tailed Hypotheses

Two-Tailed vs. One-Tailed Hypotheses

Two-Tailed Hypothesis

  • States there will be significant differences between groups, but the direction isn't specified.
  • Null hypothesis: Group 1 mean = Group 2 mean.
  • Research hypothesis: Group 1 mean ≠ Group 2 mean.
  • As the narrator explains 'We're basically saying either one of these groups can be higher, but there will be some kind of statistically significant difference. That is a two-tailed test. There's a difference; I don't know which direction'.

One-Tailed Hypothesis

  • States one group will perform significantly better or worse than the other.
  • Example: Group 1 mean > Group 2 mean.

Advantages of Each Hypothesis

  • The narrator then asks the question 'Now, I know you're wondering why would you do either/or? What is the big difference? Well, each hypothesis has its advantages'
Standard Distribution and Regions of Rejection/Non-Rejection
  • Region of Non-Rejection: If the T critical value falls within this area, you cannot reject the null hypothesis.
  • Region of Rejection: If the T critical value falls within this area, you must reject the null hypothesis.
  • T critical value: Value used to determine if groups are from the same distribution or different distributions. The narrator mentions the importance of understanding the underlying goal of this test, 'think about what we're doing: what we're really trying to say is, are these two groups from the same distribution, or are they from two different distributions?'
    Hypothetical situation: If both groups fall within the same distribution, there's a high probability they are the same (no statistically significant difference).
Two-Tailed Test and Alpha
  • Has two zones of rejection: one in the negative tail and one in the positive tail.
  • With an alpha of 0.05, the alpha is split in half for both tails (2.5% in each tail).
  • Willing to take a 2.5% chance of incorrectly rejecting the null in both tails.
  • The narrator explains 'So really, what you're saying is you're willing to take a 2.5 percent chance of incorrectly rejecting the null, where observed differences are due to sampling error in both the positive tail and the negative tail'
  • The willingness to take the risk is based on the fact that as observations fall farther from the null, you are more confident that the samples are from two different populations.
Examples
  • Population mean = 100, Sample mean = 103: Falls in the zone of non-rejection, so fail to reject the null.
  • Population mean = 100, Sample mean = 132: Falls in the region of rejection/significance, indicating a different population.
  • Population mean = 100, Sample mean = 55: Falls in the region of rejection/significance, indicating a different population.

One-Tailed Test

  • Determines if the value falls in the positive or negative tail.
  • Does not split the alpha; uses the full alpha level (e.g., 5%).
  • A 5% chance of incorrectly rejecting the null in the specified tail.
Examples:
  • Population mean = 100, Sample mean = 132: Can confidently say this sample comes from a different population.
  • Population mean = 100, Sample mean = 70: Falls within the zone of non-rejection.

Two-Tailed vs. One-Tailed: Considerations

  • If unsure which group will perform better, use a two-tailed test.
  • With a two-tailed test and alpha == 0.025 in each tail, a score of 70 would not be significant.
  • If you're unsure, you probably need to use a two-tailed test.
  • If Sample mean = 55 and population mean = 100, it falls in the zone of significance/rejection, so reject the null.