Explain what a P-value is. What is the criterion for rejecting the null hypothesis using the P-value approach?

Verified step by step guidance

1. A P-value is the probability of obtaining a test statistic at least as extreme as the one observed, assuming that the null hypothesis is true. It quantifies how compatible the observed data is with the null hypothesis.
2. To interpret the P-value, we compare it to a predetermined significance level, often denoted by \(\alpha\), which represents the threshold for deciding whether the observed data is statistically significant.
3. The criterion for rejecting the null hypothesis using the P-value approach is: if the P-value is less than or equal to the significance level \(\alpha\), we reject the null hypothesis. Mathematically, this is expressed as \(\text{P-value} \leq \alpha\).
4. If the P-value is greater than \(\alpha\), we do not have enough evidence to reject the null hypothesis, so we fail to reject it.
5. In summary, the P-value helps us decide whether the observed data is unlikely under the null hypothesis, guiding us to either reject or fail to reject the null hypothesis based on the chosen significance level.

Key Concepts

P-value

The P-value measures the probability of obtaining test results at least as extreme as the observed results, assuming the null hypothesis is true. It quantifies the evidence against the null hypothesis, with smaller values indicating stronger evidence to reject it.

Null Hypothesis

The null hypothesis is a default assumption that there is no effect or difference in the population. It serves as the starting point for statistical testing, and the goal is to determine whether there is enough evidence to reject it.

Criterion for Rejecting the Null Hypothesis

The null hypothesis is rejected if the P-value is less than a predetermined significance level (commonly 0.05). This means the observed data is unlikely under the null hypothesis, suggesting the alternative hypothesis may be true.