p-value definition and examples

📚 Quick Study Guide

• 🧪 Definition: The p-value is the probability of obtaining results as extreme as, or more extreme than, the observed results of a statistical hypothesis test, assuming that the null hypothesis is correct.
• 🔢 Significance Level: A pre-determined threshold (often 0.05) used to decide whether to reject the null hypothesis. If the p-value is less than or equal to the significance level, we reject the null hypothesis.
• 📉 Interpreting p-values:
  • 💡 Small p-value (≤ α): Strong evidence against the null hypothesis. Reject the null hypothesis.
  • 🤔 Large p-value (> α): Weak evidence against the null hypothesis. Fail to reject the null hypothesis.
• 📊 Common Scenarios:
  • 📝 Hypothesis Testing: Determining if there's a significant difference between two groups.
  • 📈 Regression Analysis: Assessing the significance of predictor variables.
• 🧮 Key Formula (Conceptual): P-value = P(observed data or more extreme | null hypothesis is true)

Practice Quiz

1. Which of the following is the best definition of a p-value?
   1. A. The probability that the null hypothesis is true.
   2. B. The probability of observing data as extreme as, or more extreme than, the observed data if the null hypothesis is true.
   3. C. The significance level of the test.
   4. D. The probability that the alternative hypothesis is true.
2. What does a small p-value (e.g., 0.01) indicate?
   1. A. Strong evidence in favor of the null hypothesis.
   2. B. Strong evidence against the null hypothesis.
   3. C. The null hypothesis is definitely true.
   4. D. The alternative hypothesis is definitely false.
3. In hypothesis testing, if the p-value is greater than the significance level (alpha), what decision do you make?
   1. A. Reject the null hypothesis.
   2. B. Fail to reject the null hypothesis.
   3. C. Accept the alternative hypothesis.
   4. D. Revise the significance level.
4. A researcher conducts a study and obtains a p-value of 0.08. If the significance level is set at 0.05, what is the appropriate conclusion?
   1. A. Reject the null hypothesis.
   2. B. Fail to reject the null hypothesis.
   3. C. Accept the null hypothesis.
   4. D. The test is inconclusive.
5. Which of the following is NOT a typical use of p-values?
   1. A. Determining the statistical significance of regression coefficients.
   2. B. Comparing the means of two groups.
   3. C. Estimating the sample size needed for a study.
   4. D. Assessing the effectiveness of a treatment.
6. What is a common significance level (alpha) used in statistical hypothesis testing?
   1. A. 0.01
   2. B. 0.05
   3. C. 0.10
   4. D. All of the above are commonly used.
7. A clinical trial is conducted to test the effectiveness of a new drug. The p-value is calculated to be 0.03. If the significance level is 0.05, what does this indicate?
   1. A. The drug is not effective.
   2. B. There is statistically significant evidence that the drug is effective.
   3. C. The null hypothesis is true.
   4. D. The alternative hypothesis is false.