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The questions in the image relate to hypothesis testing and statistical decision-making. Here's a breakdown of each:

**f. What is the decision rule using p-value?**
   - The decision rule with the p-value is: 
     - If the p-value is less than or equal to the significance level (α), reject the null hypothesis (H₀).
     - If the p-value is greater than the significance level, fail to reject the null hypothesis.
   - The significance level (α) is often set at 0.05 or 0.01, depending on the problem.

**g. What is the calculated critical value of the distribution?**
   - The critical value is a point on the distribution that serves as a cutoff for rejecting the null hypothesis. It is determined based on the chosen significance level (α) and the type of test (e.g., Z-test, t-test).
   - For example, in a two-tailed test with α = 0.05, the critical value for a Z-test would typically be ±1.96 (from the standard normal distribution).

**h. What is the formula for calculating the critical value of the distribution?**
   - The critical value depends on the type of distribution and the significance level. 
     - For a Z-test (standard normal distribution), the critical value can be found using a Z-table: \( Z_{\alpha/2} \).
     - For a t-test, the critical value can be calculated using \( t_{\alpha/2, n-1} \), where n is the sample size and \( t \) is from the t-distribution.
     - In general: 
       \[ Z_{\alpha} = \text{InverseCDF}(1 - \alpha/2) \]
     where \( \alpha \) is the significance level and InverseCDF is the cumulative distribution function for the normal distribution.

**i. What is the statistical conclusion?**
   - The statistical conclusion refers to whether you reject or fail to reject the null hypothesis based on the comparison between the p-value and the significance level or the comparison between the test statistic and the critical value. 
     - If you reject the null hypothesis, you conclude that there is enough evidence to support the alternative hypothesis.
     - If you fail to reject the null hypothesis, you conclude that there is not enough evidence to support the alternative hypothesis.

**j. What is the business conclusion?**
   - The business conclusion translates the statistical results into practical, business-relevant insights. For example, if a hypothesis test in marketing showed that a new strategy significantly increased sales, the business conclusion would suggest adopting the new strategy. On the other hand, if the result wasn't significant, the business might continue with the current approach or explore other options.

Would you like further details or explanations for any of these? Here are some related questions to explore:

1. What is the significance level, and how is it chosen?
2. How is the p-value calculated in different types of tests (e.g., Z-test, t-test)?
3. What are Type I and Type II errors in hypothesis testing?
4. How do you determine whether to use a one-tailed or two-tailed test?
5. How does sample size affect the critical value and p-value?

**Tip:** Always ensure the significance level (α) is decided before the test is conducted to avoid biases in the conclusion.

What is the decision rule using p-value? What is the calculated critical value of the distribution? What is the formula for calculating the critical value of the distribution? What is the statistical conclusion? What is the business conclusion?

Explore hypothesis testing by understanding how to use the p-value for decision-making, calculate the critical value of distributions, and draw statistical and business conclusions. This guide covers significance levels, p-values, and both Z-test and t-test calculations.

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