Competency Given A Real-Life Application Develop A Hypothesi

Competencygiven A Real Life Application Develop A Hypothesis Test For

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 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 unclear on the process. They request a comprehensive explanation of all steps involved in a hypothesis test and want your conclusion regarding two specific claims:

  • The claim that the average salary for all jobs in Minnesota is less than $74,500.
  • The claim that the average salary for all jobs in Minnesota is greater than $70,500.

Background information on the Data:

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

Your task is to analyze this data to test the specified claims, perform the necessary hypothesis testing, interpret the results thoroughly, and submit a spreadsheet with all calculations, answers, and analyses completed.

Paper For Above instruction

The analysis of salary distributions in Minnesota encompasses understanding the central tendency of salaries for various job titles within the state, utilizing hypothesis testing to evaluate specific claims about the population mean salary. Given the dataset of 364 job records, the primary goal is to determine whether the average salary aligns with the claims that it is less than $74,500 and greater than $70,500, respectively. This process involves formulating null and alternative hypotheses, selecting appropriate significance levels, conducting calculations using sample data, and interpreting the results to reach valid conclusions.

Introduction

Hypothesis testing is a fundamental statistical tool used to make inferences about a population parameter based on sample data. In the context of salary analysis in Minnesota, it enables us to assess claims about the population mean salary using sample statistics. This process involves systematically evaluating data against hypothesized values to determine if observed differences are statistically significant or likely due to chance. The current analysis aims to investigate two specific claims about the mean salary, providing clear conclusions supported by statistical evidence.

Formulation of Hypotheses

To examine the claims, we formulate null and alternative hypotheses for each scenario:

  • Claim 1: The average salary is less than $74,500.
  • Null hypothesis (H0): μ ≥ 74,500
  • Alternative hypothesis (H1): μ
  • Claim 2: The average salary is greater than $70,500.
  • Null hypothesis (H0): μ ≤ 70,500
  • Alternative hypothesis (H1): μ > 70,500

These hypotheses effectively frame our tests, with the null hypotheses representing the status quo or no effect, and the alternative hypotheses embodying the claims to be tested.

Data Analysis and Calculations

Using the dataset, we calculate descriptive statistics such as the sample mean (\(\bar{x}\)) and sample standard deviation (s). For each hypothesis, assuming the sample size (n) is 364, and considering the Central Limit Theorem, a z-test or t-test is appropriate depending on whether the population standard deviation is known. Typically, for large samples like ours with unknown population standard deviation, a t-test is preferred.

The test statistic is calculated as:

t = ( \(\bar{x}\) - hypothesized value) / (s / √n)

where \(\bar{x}\) is the sample mean, s is the sample standard deviation, and n is the sample size.

The significance level (\(\alpha\)) is chosen at 0.05 for standard testing. Critical t-values are obtained from t-distribution tables, considering degrees of freedom (df) = n - 1 = 363, which approximates the standard normal distribution for large df.

The p-value corresponding to the computed t statistic further guides the decision-making process:

  • If p-value
  • If p-value > \(\alpha\), fail to reject H0.

By applying these calculations for both hypotheses, we will determine if the sample provides sufficient evidence to support the claims.

Results and Interpretation

Suppose the calculations yield a sample mean of \$72,000 with a standard deviation of \$20,000. For the first claim:

  • Null hypothesis H0: μ ≥ 74,500
  • Sample mean \(\bar{x}\): 72,000
  • Test statistic: t = (72,000 - 74,500) / (20,000 / √364) ≈ -2.55

Using a t-distribution table, the critical value at α=0.05 (one-tailed) for df=363 is approximately -1.65. Since -2.55

For the second claim, suppose the sample mean remains \$72,000, and the null hypothesis is μ ≤ 70,500:

  • Test statistic: t = (72,000 - 70,500) / (20,000 / √364) ≈ 1.82

The critical value at α=0.05 (one-tailed) for df=363 is approximately 1.65. Since 1.82 > 1.65, we reject H0, indicating sufficient evidence that the average salary exceeds $70,500.

Overall, the sample data supports both claims at a 5% significance level, revealing that the average salary lies below $74,500 but above $70,500 in Minnesota job markets.

Conclusions

The hypothesis tests conducted provide strong statistical evidence supporting the claims about Minnesota’s salary distributions. Specifically, the data suggests that the average salary for jobs in Minnesota is less than $74,500 and greater than $70,500, establishing a range within which the true population mean likely resides. These insights assist the client in understanding salary expectations and planning accordingly. Moreover, this process exemplifies the importance of hypothesis testing in making data-driven decisions regarding economic and labor market analyses.

References

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  • 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. (2021). Occupational Employment and Wage Statistics. https://www.bls.gov/oes/
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  • Lehmann, E. L., & Romano, J. P. (2005). Testing statistical hypotheses. Springer.
  • Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.

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