The Firm You Work For Has A Poorly Thought-Out Policy

The Firm You Work For Has A Rather Poorly Thought Out Policy That A

The firm you work for has a rather poorly-thought-out policy that any estimations of the cost of a certain activity must be within a margin of error of no more than ± 10% of the estimated cost. The company does not specify a required confidence level, only that this margin must be maintained. You are working on a project with an average cost of $10,000, derived from a sample of 11 observations with a known standard deviation of $1,869. The task is to determine the highest confidence level that can be used while still ensuring the margin of error does not exceed ± 10% of the estimate, i.e., ± $1,000.

---

Paper For Above instruction

In project management and cost estimation, understanding the interplay between the margin of error and confidence levels is crucial for accurately interpreting data and making reliable predictions. The given scenario illustrates a common challenge: determining the maximum confidence level permissible under specific error margins, especially when working with small sample sizes and known standard deviations.

Understanding the Margin of Error and Confidence Level

The margin of error (ME) quantifies the range within which the true population parameter is likely to fall, given a certain confidence level. It is mathematically expressed as:

ME = z* (σ / √n)

where z* is the critical value corresponding to the selected confidence level, σ is the population standard deviation, and n is the sample size.

In this scenario, the parameters are:

  • Estimated mean (x̄): $10,000
  • Standard deviation (σ): $1,869
  • Sample size (n): 11
  • Desired maximum margin of error: ± $1,000

Calculating the Critical Z-Value

Rearranging the margin of error formula to solve for z* gives:

z = ME  (√n / σ)

Substituting the known values:

z = 1,000  (√11 / 1,869)

Calculating √11 ≈ 3.3166:

z = 1,000  (3.3166 / 1,869) ≈ 1,000 * 0.001772 ≈ 1.772

Thus, the critical z-value must not exceed approximately 1.772 to satisfy the ±10% margin of error criterion.

Relating z* to Confidence Level

The z-value corresponds to a certain confidence level in the standard normal distribution. Consulting a standard normal distribution table, a z-value of approximately 1.772 is associated with a confidence level between 96% and 97%. Specifically, the two-tailed confidence level is given by:

Confidence Level ≈ 2  P(Z ) - 1

where P(Z ) is the cumulative probability up to z.

From the normal distribution table, P(Z 

Confidence Level ≈ 2 * 0.9615 - 1 = 1.923 - 1 = 0.923 or 92.3%

However, the confidence level in terms of both tails combined is approximately 92.3%. Since the question asks for the highest confidence level, the maximum allowable confidence level is approximately 92.3%. This ensures the margin of error does not exceed ±$1,000 while respecting the company policy.

In conclusion, the highest confidence level permissible under these conditions is roughly 92%. This demonstrates how increasing confidence levels correspond to larger z-values, which in turn elevate the margin of error, potentially surpassing the allowed ±10%. Therefore, selecting a confidence level beyond approximately 92% would violate the company's error margin policy.

Implications and Recommendations

This scenario emphasizes the importance of balancing confidence levels against error margins, especially when dealing with small sample sizes and known variability. In practical applications, firms should explicitly define acceptable confidence levels for their estimates or adjust sample sizes to meet stringent error criteria. For project managers and analysts, understanding this mathematical relationship facilitates better decision-making and more accurate cost forecasting aligning with organizational policies.

References

  • Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury Press.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2012). Introduction to the Practice of Statistics (8th ed.). W.H. Freeman.
  • Ross, S. M. (2014). Introduction to Probability and Statistics for Engineers and Scientists (5th ed.). Academic Press.
  • Triola, M. F. (2018). Elementary Statistics (13th ed.). Pearson Education.
  • Wasserman, L. (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.
  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences (8th ed.). Cengage Learning.
  • Freedman, D., Pisani, R., & Purves, R. (2007). Statistics (4th ed.). W.W. Norton & Company.
  • Newbold, P., Carlson, W. L., & Thorne, B. (2013). Statistics for Business and Economics (8th ed.). Pearson.
  • Lehmann, E. L., & Casella, G. (1998). Theory of Point Estimation. Springer.
  • Rice, J. A. (2007). Mathematical Statistics and Data Analysis. Cengage Learning.

Related Assignments