State the null and alternate hypothesis State Critical Value What is the Test Statistic Value State the p-value What is your decision Write a conclusion

To answer your request thoroughly, I'll provide a general outline of how to formulate a null and alternate hypothesis, state the critical value, determine the test statistic value, and explain the p-value, final decision, and conclusion. However, I need specifics about the context or the data to provide exact numerical values. Below is a clear structure for these components:

1. State the Null and Alternate Hypothesis

• Null Hypothesis (H₀): This is usually a statement of no effect or no difference. For example, H₀: μ = μ₀ (the population mean equals a specific value).
• Alternate Hypothesis (H₁ or Hₐ): This represents the outcome we are testing for. For example, H₁: μ ≠ μ₀ (the population mean is not equal to a specific value).

2. State Critical Value

• Critical values are determined by the significance level (α), common values are 0.05, 0.01, etc.
• For a two-tailed test, you would find critical values that correspond to α/2 in each tail of the distribution (e.g., for α = 0.05, find critical z-values for 0.025 in each tail).

3. What is the Test Statistic Value

• The test statistic (e.g., z, t) is calculated based on sample data.
• For a z-test:
  $z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$ where:
  • $\bar{x}$ = sample mean
  • $\mu$ = population mean under the null hypothesis
  • $\sigma$ = population standard deviation
  • $n$ = sample size

4. State the p-value

• The p-value is calculated based on the test statistic. It indicates the probability of obtaining a result at least as extreme as the one observed, under the assumption that H₀ is true.
• For example, when using a z-test, it can be determined from z-tables or using statistical software.

5. What is your Decision

• Compare the p-value with the significance level (α):
  • If $p \leq α$: Reject H₀.
  • If $p > α$: Fail to reject H₀.

6. Write a Conclusion

• Based on the results, write a conclusion that summarizes the findings.
• Example: "At the 5% significance level, we reject the null hypothesis, suggesting evidence to support the alternative hypothesis that the population mean is significantly different from μ₀."

Final Answer

Please provide specific values or context for the hypotheses testing, including sample data, significance level, and population parameters if you need numerical results. Otherwise, the steps above outline the general process for hypothesis testing.