Null Hypothesis

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Null Hypothesis

A null hypothesis is a statistical statement that assumes no real effect or difference exists in the population being studied — that any observed variation in sample data is due to chance. It is the baseline proposition used in hypothesis testing and is denoted H0.

How it works

  • Hypothesis testing evaluates whether sample data provide enough evidence to reject H0 in favor of an alternative hypothesis (H1).
  • The null hypothesis is assumed true for the purpose of testing. The analysis determines whether the observed sample outcome is plausibly produced under H0.
  • If the sample result is unlikely under H0, we reject H0; if it is plausible, we fail to reject H0. Important: a null hypothesis can be rejected but not proven true.

Alternative hypothesis

  • The alternative hypothesis (H1) contradicts H0. If H0 states “no difference” or “no effect,” H1 states that a difference or effect exists.
  • H1 can be two-sided (e.g., mean ≠ value) or one-sided (e.g., mean > value or mean < value), depending on the research question.

Common examples

  • Fair game: H0: expected earnings = 0. H1: expected earnings ≠ 0. Repeated plays produce sample averages that are compared to zero.
  • School test scores: H0: population mean = 7. H1: population mean ≠ 7. A sample of student scores is used to decide whether the observed mean is consistent with 7.
  • Mutual fund returns: H0: mean annual return = 8%. H1: mean annual return ≠ 8%. A sample of returns is compared to the claimed long-run average.

How hypotheses are tested

Typical four-step testing process:
1. State H0 and H1 so they are mutually exclusive.
2. Choose a test and analysis plan (statistic, assumptions, significance level).
3. Compute the test statistic from the sample data.
4. Decide: reject H0 or fail to reject H0 based on the result.

Key tools:
– Test statistic (t, z, chi-square, etc.) summarizes how far the sample departs from H0.
– p-value: the probability of observing a result as extreme (or more) than the sample result if H0 is true. A small p-value suggests the sample is unlikely under H0.
– Significance level (α): a preselected threshold (commonly 0.05) below which H0 is rejected. This reflects tolerance for Type I error (false positive). Failing to reject H0 risks a Type II error (false negative).

Use in finance and investing

  • Analysts use null hypotheses to evaluate strategies, models, or market claims. For example, comparing an active strategy to buy-and-hold:
  • H0: mean return(strategy) = mean return(buy-and-hold)
  • H1: mean return(strategy) > mean return(buy-and-hold)
  • Statistical tests (and p-values) determine whether observed outperformance can be attributed to chance or is statistically significant.
  • Null hypothesis testing supports disciplined, evidence-based decisions but does not prove real-world causation; it only assesses whether observed data are improbable under H0.

Practical notes

  • Formulate H0 to reflect “no effect” or the status quo, and choose H1 to capture the alternative you want to detect.
  • Directional questions require one-sided tests; existence questions typically use two-sided tests.
  • Results depend on sample size, variability, and test assumptions — larger samples can detect smaller effects.

Bottom line

The null hypothesis is the default assumption of no effect or difference used in statistical testing. Hypothesis testing assesses whether sample data provide sufficient evidence to reject that assumption in favor of an alternative. Properly formulated hypotheses and careful interpretation of p-values and errors are essential for reliable conclusions in research and finance.