Competency Given A Real-Life Application Develop Confidence
Competencygiven A Real Life Application Develop A Confidence Interval
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. You have previously explained some of the basic statistics to your client already, and he really liked your work. Now he wants you to analyze the confidence intervals.
Background information on the Data The data set in the spreadsheet consists of 364 records from the Bureau of Labor Statistics. The data contains various job titles with annual salaries ranging from approximately $30,000 to $200,000 for Minnesota. Your task is to develop a confidence interval for the population mean salary and interpret the results based on this dataset.
Paper For Above instruction
Developing a confidence interval for a population mean salary involves understanding the characteristics of the dataset, calculating the appropriate statistical measures, and interpreting the results within the context of the data. In this case, the goal is to estimate the average salary of jobs in Minnesota within a certain confidence level, typically 95%, providing a range that likely contains the true mean salary for all applicable jobs.
The first step involves descriptive statistical analysis of the sample data. With 364 salary records, the sample mean (\(\bar{x}\)) and the sample standard deviation (s) are computed. These measures provide a summary of the central tendency and variability within the salary data. For example, suppose the sample mean salary is $85,000 with a standard deviation of $25,000. These figures serve as the basis for constructing the confidence interval.
To calculate the confidence interval, the formula for the standard error of the mean (SEM) is applied:
\[ SEM = \frac{s}{\sqrt{n}} \]
where s is the sample standard deviation and n is the sample size (364). Using the SEM, the confidence interval (CI) for the population mean is given by:
\[ \bar{x} \pm z^* \times SEM \]
Where \(z^\) corresponds to the critical value from the standard normal distribution for the desired confidence level. For a 95% confidence level, \(z^ \approx 1.96\).
Suppose the calculated SEM is approximately \$1320. Then, the confidence interval becomes:
\[ \$85,000 \pm 1.96 \times \$1320 \]
which provides a range from about \$82,412 to \$87,588. This means we are 95% confident that the true average salary for jobs in Minnesota falls within this interval.
Interpreting the confidence interval, it’s important to recognize that it reflects the uncertainty inherent in sampling. It does not guarantee that the true mean is within this range, but rather that, in repeated sampling under similar conditions, approximately 95% of such intervals will contain the true population mean.
In practical terms, this interval can assist the client in understanding salary expectations within Minnesota and making informed decisions based on seasonal or industry trends, as well as setting competitive salary benchmarks or evaluating market conditions.
Limitations of the analysis include the assumption that the sample is representative of the entire population of job salaries in Minnesota and that the data distribution is roughly normal or sufficiently large for the Central Limit Theorem to hold. Since the sample size is quite large (364 records), the normal approximation is justified, increasing confidence in the results.
In conclusion, constructing a confidence interval for the mean salary provides valuable insights into the salary landscape for Minnesota jobs. The process involves calculating descriptive statistics, applying the appropriate formulas for standard error, and interpreting the results within the context of the data and the client's needs. These statistical tools are essential for making data-driven decisions in a business environment.
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