This Activity Is A Practice In Preliminary Understanding Of

This activity is a practice in preliminary understanding of confidence interval estimation

This activity is a practice in preliminary understanding of confidence interval estimation. Please read the following scenario and respond to the questions related to the scenario: You want to estimate the production days that would be lost during the next three months by sampling the vacation intentions of a sample of employees. You randomly select 36 employees in the organization and find that the average number of days they intend taking off is 16 days during the coming three summer months, with a standard deviation of seven (7) days. Based on these sample statistics, you want to estimate at a 99% confidence level, the days that will be lost because of the entire population of workers taking vacation time during the next three months, so that the plant manager knows how much temporary help he should plan on hiring during the summer months in order for work to proceed smoothly.

Assume that days of intending to take vacation by an employee is independent of all other employees’ vacation intentions.

Questions:

  1. Construct the confidence interval for the mean vacation time of an employee based on the above sample results.
  2. If there are 100 employees in the organization expected to take vacation during summer, what is the maximum (most pessimistic) and minimum (most optimistic) number of days of labor that temporary help would be needed for production to proceed smoothly?
  3. Based on the same sample results, how can you reduce the interval estimated in number 2 above? Please, provide an example and calculate the new confidence interval. What would be a trade-off for reducing the confidence interval length?

Paper For Above instruction

Estimating the expected vacation days during the summer months is crucial for workforce planning and maintaining production efficiency in an organization. Confidence interval estimation provides a statistical method to predict the range within which the true average vacation days of the entire employee population is likely to fall, based on sample data. This essay discusses the construction of confidence intervals, implications for staffing, and methods to refine these estimates for better decision-making.

To construct a confidence interval for the mean vacation time based on the sample, we rely on the sample mean, standard deviation, sample size, and the desired confidence level. Given the sample mean (16 days), standard deviation (7 days), and the sample size (36 employees), we first identify the appropriate critical value from the t-distribution because the population standard deviation is unknown and the sample size is relatively small. For a 99% confidence level and degrees of freedom (df) = 35, the t-value approximates to 2.719 (Tabachnick & Fidell, 2013).

The formula for the confidence interval is:

CI = sample mean ± tα/2 * (s / √n)

where s is the sample standard deviation, n is the sample size, and tα/2 is the critical t-value for desired confidence.

Substituting the values:

CI = 16 ± 2.719 * (7 / √36)

CI = 16 ± 2.719 * (7 / 6)

CI = 16 ± 2.719 * 1.167

CI = 16 ± 3.177

Thus, the 99% confidence interval is approximately (12.823 days, 19.177 days). This interval suggests that we can be 99% confident the true average vacation days per employee during the summer fall within this range.

Regarding staffing needs, if 100 employees are expected to take summer vacation, the minimum total vacation days is calculated by assuming the lower limit of the confidence interval, while the maximum is based on the upper limit:

  • Minimum total vacation days: 100 × 12.823 ≈ 1,282 days
  • Maximum total vacation days: 100 × 19.177 ≈ 1,918 days

Therefore, the plant manager should plan for anywhere between approximately 1,282 and 1,918 days of production downtime, which can be mitigated with appropriate temporary staffing strategies.

Reducing the confidence interval's length improves the precision of the estimate. One way to achieve this is by increasing the sample size. For example, increasing the sample size from 36 to 100 employees while maintaining a similar standard deviation reduces the standard error, thus narrowing the interval. If we assume the sample standard deviation remains at 7 days, then for n = 100:

Standard error: s / √n = 7 / √100 = 7 / 10 = 0.7

Using the same critical t-value for a 99% confidence level with df = 99, approximately 2.626:

Confidence interval: 16 ± 2.626 * 0.7 = 16 ± 1.838

This yields an interval of approximately (14.162 days, 17.838 days), which is narrower than the previous estimate.

The trade-off for reducing the confidence interval length is the need for a larger sample size, which entails more time and resources for data collection. Additionally, increasing the sample size may not always be feasible due to organizational constraints, making it necessary to balance the desire for precision with practical limitations.

References

  • Tabachnick, B. G., & Fidell, L. S. (2013). Using Multivariate Statistics (6th ed.). Pearson.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics (7th ed.). W. H. Freeman.
  • Everitt, B. S. (2002). The Cambridge Dictionary of Statistics. Cambridge University Press.
  • Textbook of Statistics: Principles, Methods, and Applications. (2016). Academic Press.
  • Creswell, J. W. (2014). Research Design: Qualitative, Quantitative, and Mixed Methods Approaches. Sage Publications.
  • Newbold, P., Carlson, W. L., & Thorne, B. (2013). Statistics for Business and Economics. Pearson.
  • Glen, S. (2015). Principles of Sampling. Statistics How To. https://www.statisticshowto.com
  • Siegel, S., & Castellan, N. J. (1988). Nonparametric Statistics for the Behavioral Sciences. McGraw-Hill.
  • Agresti, A., & Franklin, C. (2013). Statistical Methods for the Social Sciences. Pearson.
  • Zimmerman, D. W. (1997). The Treatment of Outliers. Review of Educational Research, 67(2), 219-262.

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