Statistical Significance Examples in Statistics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Statistical Significance.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept Recap

A result is statistically significant when the p-value falls below a predetermined threshold (alpha, typically 0.05), indicating that the observed effect is unlikely to have occurred by random chance alone. Statistical significance is a binary decision criterion used in hypothesis testing — it does not measure the size or practical importance of the effect.

Statistical significance is a decision rule: before looking at data, you set a threshold (usually 5%). If your p-value is below this threshold, you declare the result 'significant' - meaning unlikely to be just random noise. It's not about importance; it's about confidence that something real is happening.

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How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Statistical significance means the p-value is below the chosen threshold (alpha), suggesting the result is unlikely due to chance — not that it is practically important.

Common stuck point: Statistical significance is not the same as practical importance. A tiny, meaningless difference can be statistically significant with a large enough sample size.

Sense of Study hint: To determine statistical significance, compare your p-value to the chosen alpha level (usually 0.05). If p < \alpha, the result is statistically significant and you reject the null hypothesis. If p \geq \alpha, you fail to reject. Always report the actual p-value alongside the significance decision, and consider effect size to judge practical importance.

Worked Examples

Example 1

hard
A drug trial finds a statistically significant reduction in blood pressure (p = 0.02). The mean reduction was 2 mmHg. Is this result practically significant?

Solution

  1. 1
    Step 1: Statistical significance (p = 0.02 < 0.05) means the reduction is unlikely due to chance alone.
  2. 2
    Step 2: However, a 2 mmHg reduction is very small clinically — most doctors would not consider this meaningful.
  3. 3
    Step 3: Statistical significance does not imply practical significance. Large samples can detect tiny differences that have no real-world importance.

Answer

Statistically significant but not practically significant — the effect size is too small to be clinically meaningful.
Statistical significance tells us whether an effect exists; practical significance tells us whether the effect matters. Both should be considered when interpreting results.

Example 2

hard
Explain the difference between a Type I error and a Type II error in hypothesis testing.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

hard
A study with 10,000 participants finds a statistically significant difference in test scores between two teaching methods (p = 0.001), but the difference is only 0.5 points out of 100. Discuss.

Example 2

hard
A treatment improves average test scores by 12 points, but the p-value is 0.08. At \alpha = 0.05 is the result statistically significant, and could the effect still be practically important?
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Background Knowledge

These ideas may be useful before you work through the harder examples.

p valuehypothesis testing