P-value

Definition

The p-value is the probability of obtaining a test statistic at least as extreme as the one actually observed, assuming the null hypothesis is true. It is a crucial concept in hypothesis testing that helps determine the statistical significance of a result.

5 Must Know Facts For Your Next Test

1. The p-value is used to determine the statistical significance of a test result in hypothesis testing.
2. A smaller p-value indicates stronger evidence against the null hypothesis, suggesting the observed result is unlikely to have occurred by chance.
3. The p-value is compared to the predetermined significance level (α) to decide whether to reject or fail to reject the null hypothesis.
4. The p-value is affected by the sample size, effect size, and variability in the data, with larger samples and larger effects generally resulting in smaller p-values.
5. The p-value is a continuous measure of the strength of evidence against the null hypothesis, not a binary decision of 'significant' or 'not significant.'

Review Questions

• Explain how the p-value is used in the context of contingency tables (3.4) to determine the statistical significance of the relationship between two categorical variables.
  • In the context of contingency tables (3.4), the p-value is used to determine the statistical significance of the relationship between two categorical variables. The null hypothesis typically states that the two variables are independent, meaning there is no association between them. The test statistic, such as the chi-square statistic, is calculated from the observed and expected frequencies in the contingency table. The p-value represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. If the p-value is less than the predetermined significance level (e.g., α = 0.05), the null hypothesis is rejected, indicating a statistically significant relationship between the two variables. The p-value, in this case, helps quantify the strength of evidence against the independence of the variables.
• Describe how the p-value is used in the context of the Central Limit Theorem (7.4) to assess the statistical significance of the difference between a sample mean and a population mean.
  • In the context of the Central Limit Theorem (7.4), the p-value is used to assess the statistical significance of the difference between a sample mean and a population mean. The null hypothesis typically states that the sample mean is equal to the population mean, and the alternative hypothesis states that the sample mean is different from the population mean. The test statistic, such as the z-score or t-statistic, is calculated based on the sample mean, population mean, and the standard error of the mean. The p-value represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. If the p-value is less than the predetermined significance level (e.g., α = 0.05), the null hypothesis is rejected, indicating a statistically significant difference between the sample mean and the population mean. The p-value, in this case, helps determine the likelihood that the observed difference is due to chance or is a true reflection of a difference in the population.
• Explain how the p-value is used in the context of hypothesis testing (9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 10.1, 10.2, 10.3, 10.4, 10.5) to make decisions about the null and alternative hypotheses.
  • In the context of hypothesis testing (9.1, 9.2, 9.3, 9.4, 9.5, 9.6, 10.1, 10.2, 10.3, 10.4, 10.5), the p-value is a crucial component in the decision-making process. The null hypothesis typically states that there is no significant difference or relationship between the variables being tested, while the alternative hypothesis states that there is a significant difference or relationship. The test statistic is calculated based on the sample data, and the p-value represents the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true. If the p-value is less than the predetermined significance level (e.g., α = 0.05), the null hypothesis is rejected, indicating that the observed result is statistically significant and unlikely to have occurred by chance. Conversely, if the p-value is greater than the significance level, the null hypothesis is not rejected, suggesting that the observed result is not statistically significant. The p-value, in this case, helps determine whether the evidence from the sample data is strong enough to conclude that the null hypothesis is false and the alternative hypothesis is true.

Related terms

Null Hypothesis: The null hypothesis is a statement that there is no significant difference or relationship between the variables being tested.

Alternative Hypothesis:

The alternative hypothesis is a statement that there is a significant difference or relationship between the variables being tested.

Significance Level:

The significance level, denoted as α, is the probability of rejecting the null hypothesis when it is actually true, also known as a Type I error.