Meaning of P-value and Confidence Intervals

Core Question of Statistical Reasoning: The primary question asked when analyzing data relative to a hypothesis is: "Are the data surprising, given the null hypothesis?"
Calculating Probability: reasoning is based on determining exactly how likely the observed data would be if the null hypothesis ( $H_0$ ) were a true model of the world.
Formal Definition of a P-value: A P-value is the probability of observing data like the current sample (or something even less likely/more extreme) specifically under the assumption that the null hypothesis is true.
The Role of the Null Hypothesis ( $H_0$ ): The null hypothesis is the entity that "builds" the distribution or the model used for testing. It serves as the center of the model against which the sample statistic is measured.

Threshold Context: Annotations suggest that a high P-value is often considered to be $5\%$ or higher.
Consistency with the Model: When a P-value is high, it indicates that the observed events have a high probability of occurring naturally. Therefore, the data are consistent with the model proposed by the null hypothesis.
Formal Conclusion: "Fail to Reject": - If the P-value is high, there is no evidence against the null hypothesis. - This results in the formal decision to "fail to reject" the null hypothesis.
Limitations of the Conclusion: - Failing to reject the null hypothesis is not a proof that the null hypothesis is true. - Many other similar hypotheses could potentially account for the observed data. - The most that can be stated is that the null hypothesis "doesn't appear to be false." - This is considered a "weak conclusion," but it is the only one statistically entitled to the researcher.

Statistical Significance: A low P-value creates what is known as "statistical significance."
Evidence of Rare Events: A low P-value indicates that it is very unlikely the researcher would observe these specific data points if the null hypothesis were true.
Conflict Between Model and Data: A low P-value puts the model ( $H_0$ ) and the data at odds. This presents a choice: - Option A: The null hypothesis is correct, and the researcher has witnessed a truly "remarkable" or rare event. - Option B: The null hypothesis is wrong, and the model used to compute the P-value was incorrect.
Decision to Reject: If a researcher prioritizes data over initial assumptions, a low P-value leads them to reject the null hypothesis. This suggests that a different model may be correct, making the data appear less remarkable.

The "Surprise" Factor: The P-value measures precisely how "surprised" a statistician is by the data.
Distance from the Center: - The calculation essentially asks: "Is my sample statistic ( $\hat{p}$ ) far from the center ( $H_0$ )?" - If the P-value is significantly far from the center of the $H_0$ distribution, the model is likely wrong, leading to a rejection of $H_0$ . - A P-value approaching $0\%$ indicates the data is very far from the null hypothesis center.
Summary of Logic: - Large P-value: Sample is consistent with $H_0$ . - Small P-value: Sample is surprising/remarkable given $H_0$ , leading to a rejection of the null.