Inferential Statistics Notes

Inferential Statistics

  • Steps:
    • Translate the research question into hypotheses (research and null).
    • Conduct the study.
    • Analyze results using statistical analysis.
    • Determine whether to reject or retain the null hypothesis.
    • Note: You never accept the null hypothesis.

Null Hypothesis

  • Notation: H_0 (H with a subscript zero, derived from 'no').
  • Definition: A hypothesis that can be rejected or retained.
  • It is the opposite of the research hypothesis.
  • Example: If you believe a pill influences blood pressure, the null hypothesis is that there will be no group mean differences between the group that receives the pill and the group that does not receive the pill.
  • The null hypothesis posits that any difference between groups is due to mere chance.
  • Even if two groups both receive the pill, their mean blood pressures won't be exactly the same, but the difference might be small enough to attribute to individual variations or chance.

Research Hypothesis (Alternative Hypothesis)

  • Notation: H_1
  • Definition: States what will happen if the null hypothesis is rejected.
  • Example: There will be a statistically significant difference between groups' blood pressure, generally based upon the intervention.
  • While we can't definitively say the difference is only related to the intervention, rejecting the null suggests the intervention has an effect.

Equation Form of Hypotheses

  • Null Hypothesis: H_0: \text{Mean of Group 1} = \text{Mean of Group 2}
  • Research Hypothesis:
    • H_1: \text{Mean of Group 1} \neq \text{Mean of Group 2} (two-tailed test)
    • H_1: \text{Mean of Group 1} > \text{Mean of Group 2} (one-tailed test)
    • H_1: \text{Mean of Group 2} > \text{Mean of Group 1} (one-tailed test)

Examples

  • Null Hypothesis: The means for sample 1 and sample 2 are equal.
  • Research Hypothesis (Two-Tailed): The means for sample 1 and sample 2 are not equal.
  • Null Hypothesis: Group means based on intervention A versus B are not statistically significant.
  • Research Hypothesis: There is a significant mean difference between groups 1 and 2 due to the intervention.

Hypothesis Testing

  • The null hypothesis is always tested.
  • Goal: Determine whether to reject or retain the null hypothesis.
  • Rejecting the null hypothesis: Supports the research hypothesis.
  • Maintaining the null hypothesis: Research hypothesis cannot be supported.

Why You Never "Accept" the Null Hypothesis

  • The null hypothesis is a probability-based statement regarding chance.
  • You can never state that a difference is absolutely due to chance because a true effect might exist but was missed.
  • Reasons for potentially missing a real effect:
    • Bad sample.
    • Sample size is too small.
  • Sampling is never perfect, even with random sampling. It may not yield a perfectly representative sample.
    • Example: Randomly sampling from a phone book might yield more females than males, reflecting the population, but it's not guaranteed.
    • Example: Flipping a coin 100 times could theoretically result in 100 heads, though unlikely.
  • Biases in research procedures can also lead to incorrect conclusions of no statistically significant difference when a true difference exists.
  • Research studies are not perfect, just as samples are not perfect.