how to know when to be reasonably confident or very confident when accepting the alternative hypothesis

In statistical hypothesis testing, determining when to be **reasonably confident** or **very confident** in accepting the **alternative hypothesis** (denoted as \( H_a \)) depends largely on the **p-value** and the **significance level** (denoted as \( \alpha \)).

Here's a breakdown of how confidence relates to hypothesis testing:

### 1. **P-value Interpretation**
The p-value indicates the probability of observing the test results (or more extreme) given that the null hypothesis (\( H_0 \)) is true. Lower p-values suggest stronger evidence against the null hypothesis and in favor of the alternative hypothesis.

- **Reasonably Confident**: If the p-value is **less than or equal to** your chosen significance level (\( \alpha \)), you reject the null hypothesis. For many situations, a p-value of **0.05** or smaller would be considered reasonable evidence to accept the alternative hypothesis.
- **Very Confident**: If the p-value is **significantly smaller** than your chosen significance level, such as **0.01 or 0.001**, this indicates stronger evidence against the null hypothesis, and you would be very confident in rejecting \( H_0 \) in favor of \( H_a \).

### 2. **Significance Level (α)**
The significance level (\( \alpha \)) is a threshold you set before conducting the test to determine the likelihood of rejecting \( H_0 \) when it is actually true (a type I error). Commonly, **α = 0.05** is used, but you can adjust this depending on the context:

- **Reasonably Confident**: A p-value less than or equal to \( \alpha = 0.05 \) typically means you can reject \( H_0 \) with reasonable confidence.
- **Very Confident**: If the p-value is less than a smaller threshold, such as **0.01** or **0.001**, you would be even more confident that the alternative hypothesis is true.

### 3. **Effect Size**
Confidence in the alternative hypothesis also depends on the **effect size**, which measures the magnitude of the difference or effect observed in your data. Larger effect sizes generally increase confidence in the alternative hypothesis, especially if the p-value is small.

### 4. **Confidence Intervals**
If you compute a confidence interval (CI) for a parameter (like a mean or proportion), and the interval does **not** include the null hypothesis value (e.g., 0 for differences, 1 for ratios), this can strengthen your confidence in rejecting the null hypothesis. The narrower the CI and the further it is from the null hypothesis value, the stronger your confidence.

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### Summary

- **Reasonably Confident**: You reject \( H_0 \) if p-value \( \leq 0.05 \), and the effect size is moderate.
- **Very Confident**: You reject \( H_0 \) if p-value \( \leq 0.01 \) or \( 0.001 \), with a large effect size and narrow confidence interval.

#### Do you need further details or clarification on any specific aspect of this process? 

Here are 5 related questions:
1. What is the role of the power of a test in deciding confidence in the alternative hypothesis?
2. How do you select an appropriate significance level (\( \alpha \)) for a hypothesis test?
3. What are common mistakes made when interpreting p-values?
4. How do you calculate effect size and how does it affect confidence in rejecting the null hypothesis?
5. How do confidence intervals relate to hypothesis testing and the decision to accept or reject the null hypothesis?

**Tip**: The lower the p-value, the stronger the evidence against \( H_0 \), but always consider the context and possible consequences of making a type I error (false positive).

Learn when to be reasonably or very confident in accepting the alternative hypothesis in statistical hypothesis testing. This guide explains p-values, significance levels, effect sizes, and confidence intervals with examples for better understanding. Suitable for undergraduate students in statistics.

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