Describe The 8 Steps In The Process For Hypothesis Testing

Describe the 8 steps in the process for hypothesis testing

Explain the decision criteria for rejecting the null hypothesis for both the p-value method and the critical value method. Answer and Explanation: Enter your step-by-step answer and explanations here. The remaining problems refer to the following scenario: A claim is made that the average salary for all jobs in Minnesota is less than $75,000. You are going to test the claim using and assume that your data is normally distributed and the population standard deviation is not known.

Paper For Above instruction

Hypothesis testing is a fundamental statistical procedure used to determine whether there is enough evidence in a sample of data to infer that a certain condition holds true for the entire population. The process involves several systematic steps that ensure the test's validity and reliability. The eight-step process for hypothesis testing can be summarized as follows:

1. State the Null and Alternative Hypotheses

The first step involves formulating the null hypothesis (H₀), which represents the default or status quo assumption, and the alternative hypothesis (H₁ or Ha), which is what the researcher seeks to support. For the scenario where the claim is that the average salary in Minnesota is less than $75,000, the hypotheses are:

  • Null hypothesis (H₀): μ ≥ 75,000 (the mean salary is at least $75,000)
  • Alternative hypothesis (H₁): μ

This constitutes a left-tailed test because the alternative hypothesis tests for a mean less than a specific value.

2. Set the Significance Level (α)

The significance level, typically denoted as α, is the threshold probability for rejecting H₀. Common choices include 0.05 or 0.01, representing 5% or 1% risk of Type I error (incorrectly rejecting a true null hypothesis). Selecting α depends on the context and desired confidence level.

3. Collect Data and Compute the Sample Statistic

Gather a representative sample of salary data from Minnesota jobs. Calculate the sample mean (x̄) and sample standard deviation (s). Since the population standard deviation is not known, the sample standard deviation is used to estimate variability.

4. Determine the Test Statistic

Given that the population standard deviation is unknown and the sample size is typically small, the appropriate test statistic is the t-statistic:

t = (x̄ - μ₀) / (s / √n)

where μ₀ is the hypothesized mean ($75,000), s is the sample standard deviation, and n is the sample size.

Calculate this value using the collected data.

5. Establish the Critical Value and Rejection Region

Find the critical t-value associated with α and degrees of freedom (n - 1) from a t-distribution table. The rejection region comprises all t-values less than this critical value (for a left-tailed test). If the calculated t-statistic falls into this region, then H₀ is rejected.

6. Make a Decision

Compare the calculated t-statistic to the critical value:

  • If t
  • If t ≥ critical value, do not reject H₀.

Restate the decision in non-technical terms, such as "There is sufficient evidence to support the claim that the average salary in Minnesota is less than $75,000" or "Insufficient evidence exists to support the claim."

7. Calculate the p-value and Draw Conclusions

The p-value is the probability, under the null hypothesis, of obtaining a t-statistic as extreme or more extreme than the observed one. Calculate this p-value using software or t-distribution tables:

  • If p-value
  • If p-value ≥ α, do not reject H₀.

Compare this conclusion with the decision from the critical value approach to verify consistency. If both methods lead to the same conclusion, confidence in the results is increased.

8. Finalize and Report Findings

Summarize the results, specify whether the hypothesis was rejected or not, and interpret the findings in the context of the original claim. Discuss implications for the claim that the average salary is less than $75,000 in Minnesota based on the statistical evidence.

Decision Criteria for Rejection

P-value Method

The null hypothesis (H₀) is rejected if the p-value is less than the predetermined significance level α (e.g., 0.05). This indicates that the observed data is unlikely under H₀, supporting the alternative hypothesis.

Critical Value Method

H₀ is rejected if the test statistic falls into the rejection region, which is characterized by having a t-value less than the critical t-value for a left-tailed test at the specified α level. If the test statistic is in this region, we reject H₀ because the data is inconsistent with the null hypothesis.

Conclusion

Both the p-value and critical value approaches provide methods to decide whether to reject the null hypothesis. They are mathematically linked; a small p-value corresponds to a test statistic in the rejection region. The choice of method depends on preference and available tools, but both lead to consistent conclusions when correctly applied.

References

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