Develop A Hypothesis Test For A Real-Life Application

Develop A Hypothesis Test For A Real Life Application

Given a real-life application, develop a hypothesis test for a population parameter and its interpretation.

Scenario (information repeated for deliverable 01, 03, and 04):

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 (attached) that contains the following information: a listing of the jobs by title and the salary (in dollars) for each job. In prior engagements, you have already explained to your client about the basic statistics and discussed the importance of constructing confidence intervals for the population mean.

Your client recalls some information about hypothesis testing but is unsure about the process. They ask you to give a full explanation of all steps involved in conducting a hypothesis test and to provide a conclusion regarding the claim that the average salary for all jobs in Minnesota is less than $75,000.

Background information on the data:

The dataset in the spreadsheet contains 364 records obtained from the Bureau of Labor Statistics. It includes various job titles with annual salaries ranging approximately from $30,000 to $200,000 for Minnesota.

What to Submit:

You are required to submit the spreadsheet with all the calculations completed. Your research, analysis, and the step-by-step explanation of hypothesis testing should be included within the answers provided on the spreadsheet.

Paper For Above instruction

Introduction

Hypothesis testing is a fundamental technique in statistics used to draw inferences about a population parameter based on sample data. In this analysis, we are examining whether the average salary of jobs in Minnesota differs from a specified value, particularly testing whether the mean salary is less than $75,000. Given the dataset of 364 job salaries, hypothesis testing provides a structured approach to assess this claim statistically, ensuring decisions are grounded in data with quantifiable confidence levels.

Background and Motivation

The client’s interest in Minnesota's salary distributions necessitates rigorous statistical analysis. Employees, employers, and policymakers rely on such studies to make informed decisions, whether about salary negotiations, policy formulation, or economic forecasting. Given the range of salaries from $30,000 to $200,000, understanding whether the mean salary falls below a critical threshold (such as $75,000) can have significant implications. Hypothesis testing offers a formal framework to assess such claims objectively, moving beyond anecdotal or superficial observations.

Formulation of Hypotheses

The central statistical question involves testing a claim regarding the population mean salary (μ). The null hypothesis (H₀) posits that the average salary is equal to or greater than $75,000, while the alternative hypothesis (H₁) asserts that the average salary is less than $75,000. Formally, these hypotheses are expressed as:

  • H₀: μ ≥ 75,000
  • H₁: μ

This setup indicates a left-tailed test, as we are testing whether the mean salary is significantly less than $75,000.

Data Analysis and Calculation

Using the data provided, the first step involves calculating the sample mean (x̄) and sample standard deviation (s). Given the 364 salary entries, these statistics can be computed directly from the dataset. Suppose the sample mean is approximately $X,000 and the sample standard deviation is $Y,000 (values to be calculated from the spreadsheet). We then determine the standard error of the mean (SE) as:

SE = s / √n

where n = 364. This standard error represents the variability of the sample mean as an estimate of the population mean.

Test Statistic Computation

The test involves calculating a t-statistic to compare the sample mean against the hypothesized population mean of $75,000. The formula is:

t = (x̄ - μ₀) / (s / √n)

where μ₀ = 75,000. This t-value measures how many standard errors the sample mean is away from the hypothesized mean under the null hypothesis.

Decision Rule and Significance Level

Choosing a significance level (α) of 0.05, the critical t-value for a one-tailed test with degrees of freedom df = n - 1 = 363 is obtained from the t-distribution table or software. If the calculated t is less than the critical t-value, we reject H₀, suggesting evidence in favor of the claim that the mean salary is less than $75,000.

Interpretation of Results

If the test statistic falls into the rejection region, we conclude that there is sufficient statistical evidence to support the client's claim. Conversely, if the test statistic does not fall into the rejection region, we fail to reject H₀ and conclude that there is not enough evidence to assert that the mean salary is below $75,000.

Sample Calculation (Hypothetical)

For illustration, assume the sample mean (x̄) is $80,000 and the sample standard deviation (s) is $50,000. The standard error (SE) then is:

SE = 50,000 / √364 ≈ 2,625

The t-statistic becomes:

t = (80,000 - 75,000) / 2,625 ≈ 1.905

With degrees of freedom of 363, the critical t-value for α = 0.05 (one-tailed) is approximately -1.65. Since 1.905 is greater than -1.65, we do not reject the null hypothesis, indicating insufficient evidence to conclude that the mean salary in Minnesota is less than $75,000. However, actual results depend on the precise data calculations performed on the spreadsheet.

Conclusion

The hypothesis testing procedure provides an objective framework to evaluate the claim concerning average salaries. If the computed t-value falls into the rejection region, the client can be statistically confident that the mean salary is genuinely less than $75,000. If not, the data does not provide sufficient evidence to support the claim. This approach exemplifies how data-driven decisions can be made with quantifiable confidence, enhancing credibility and clarity in business analytics.

References

  • Craft, M. (2019). Introduction to Statistics. Pearson Education.
  • Glen, S. (2019). Hypothesis testing. Statistics How To. https://www.statisticshowto.com/probability-and-statistics/hypothesis-testing/
  • Newbold, P., Carlson, W. L., & Thorne, B. (2013). Statistics for Business and Economics. Pearson.
  • Levine, D. M., Krehbiel, T. C., & Berenson, M. L. (2019). Business Statistics: A First Course. Pearson.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2017). Introduction to the Practice of Statistics. W. H. Freeman.
  • U.S. Bureau of Labor Statistics. (2023). Occupational Employment and Wages. https://www.bls.gov/oes/
  • Rumsey, D. J. (2019). Statistics For Dummies. Wiley.
  • Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineering and the Sciences. Pearson.
  • Agresti, A., & Finlay, B. (2009). Statistical Methods for the Social Sciences. Pearson.
  • Schwab, J. (2017). Business Statistics: Communicating with Numbers. McGraw-Hill Education.

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