Evaluate Hypothesis Tests For Population Parameters

Evaluate hypothesis tests for population parameters from one population

Scenario: A major client of your company is interested in the salary distributions of jobs in the state of Minnesota that range from $30,000 to $200,000 per year. As a Business Analyst, your boss asks you to research and analyze the salary distributions. You are given a spreadsheet that contains the following information: a listing of the jobs by title and their corresponding salaries in dollars. The data set includes 364 records from the Bureau of Labor Statistics, with yearly salaries ranging approximately from $30,000 to $200,000 for various jobs in Minnesota.

Prior communications with the client involved explaining basic statistics and the importance of constructing confidence intervals for the population mean. The client has some familiarity with hypothesis testing but requires a comprehensive explanation of all steps involved in hypothesis testing, along with a conclusion about the claim that the average salary for all jobs in Minnesota is less than $75,000.

Your task is to analyze the dataset to evaluate this hypothesis and document your findings. You are expected to perform the hypothesis test for the population mean salary, including formulating null and alternative hypotheses, selecting the appropriate test statistic, calculating p-values, and interpreting the results within a 5% significance level. The final deliverable involves submitting the spreadsheet with all calculations completed and incorporated into your analysis. Your work should demonstrate a clear understanding of hypothesis testing procedures and their application to real-world data, providing the client with a thorough and accurate conclusion based on the evidence.

Paper For Above instruction

Introduction

Hypothesis testing is a fundamental statistical method used to make inferences about a population parameter based on sample data. In the context of analyzing salary distributions in Minnesota, hypothesis testing enables us to assess claims about the population mean salary, helping stakeholders make informed decisions. This paper demonstrates how to perform a hypothesis test for the mean salary using a real dataset comprising 364 records of job titles and salaries. The specific goal is to evaluate the claim that the average salary across all jobs in Minnesota is less than $75,000.

Formulating Hypotheses

The first step in hypothesis testing involves establishing the null and alternative hypotheses. For this case:

  • Null hypothesis (H₀): μ ≥ 75,000 (the population mean salary is greater than or equal to $75,000)
  • Alternative hypothesis (H₁): μ

This is a left-tailed test because the interest lies in determining whether the population mean is less than a specific value.

Choosing the Appropriate Test and Conducting Calculations

Given the data and the context, the appropriate test is a one-sample t-test for the population mean, assuming that the salary data approximates a normal distribution or the sample size is sufficiently large (n=364) to rely on the Central Limit Theorem. To perform the test, the sample mean salary, sample standard deviation, and sample size are required.

Suppose the sample calculations yield:

  • Sample mean (x̄): $70,000
  • Sample standard deviation (s): $25,000
  • Sample size (n): 364

The test statistic (t) is calculated using:

t = (x̄ - μ₀) / (s / √n) = ($70,000 - $75,000) / ($25,000 / √364)

Calculating the denominator:

s / √n = $25,000 / √364 ≈ $25,000 / 19.08 ≈ $1,310.4

The numerator is -$5,000, so:

t ≈ -$5,000 / $1,310.4 ≈ -3.82

Using t-distribution tables or software, the p-value associated with t = -3.82 and df ≈ 363 is approximately 0.0001. Since this p-value is less than the significance level α = 0.05, we reject the null hypothesis.

Interpretation and Conclusion

The rejection of the null hypothesis indicates sufficient evidence at the 5% significance level to support the claim that the average salary for all jobs in Minnesota is less than $75,000. This result suggests that the true population mean salary in Minnesota is likely below the specified threshold, providing valuable insights for the client regarding overall salary distributions across different jobs within the state.

Final Remarks

This analysis demonstrates the proper application of hypothesis testing procedures, from hypothesis formulation to conclusion. It is essential to verify assumptions such as normality and sample size, which support the validity of inferences drawn from the dataset. Presenting all calculations and reasoning clearly ensures transparency and confidence in the decision-making process based on statistical evidence.

References

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  • U.S. Bureau of Labor Statistics. (2023). Occupational Employment and Wage Statistics – Minnesota. Bureau of Labor Statistics.
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