Project Description: Study Shows Data For Delivery Times ✓ Solved

Project Description: The study shows data for delivery times

Project Description: The study shows data for delivery times of international shipments for a logistics company. The company promotes international delivery as a service that takes up to 7 days. After complaints that delivery time exceeds 7 days, the quality assurance team collected a random sample to analyze delivery times and decide if the promised delivery time should be adjusted. Assume delivery times are normally distributed and the sample is randomly selected.

Tasks:

  1. From the sample data, calculate the sample mean, sample standard deviation, and sample size.
  2. Determine the appropriate probability table to use for constructing confidence intervals for the population mean.
  3. Using Excel functions (e.g., T.INV.2T and CONFIDENCE.T), construct a 90% confidence interval for the population mean: find the t-score, margin of error, lower and upper limits, and the interval width.
  4. Construct a 99% confidence interval similarly: t-score, margin of error, lower and upper limits, and interval width.
  5. Compare the widths of the 90% and 99% confidence intervals and explain the relationship.
  6. Based on each confidence interval, state whether the team should recommend changing the promised 7-day delivery time.
  7. Describe options the team can use to reduce the margin of error for the 99% confidence interval.

Paper For Above Instructions

Introduction and goals

The objective is to analyze sampled delivery-time data to estimate the population mean delivery time and to determine whether the company’s advertised maximum of 7 days remains defensible. The required outputs are sample summary statistics (mean, standard deviation, sample size), construction of 90% and 99% confidence intervals for the population mean using the student’s t-distribution, comparison of interval widths, recommendations as to whether the 7-day promise should be changed, and strategies to reduce the 99% margin of error. The following sections outline the computation steps, interpretation criteria, and practical recommendations.

Step 1 — Sample summary statistics

Begin by computing three statistics from the sample: the sample mean (x̄), the sample standard deviation (s), and the sample size (n). In Excel these are computed with built-in functions: =AVERAGE(range), =STDEV.S(range) for sample standard deviation, and =COUNT(range) for n. These summary statistics are the basis for the confidence interval formula (Moore, McCabe, & Craig, 2014).

Step 2 — Choice of probability table (t versus z)

Because the population standard deviation is unknown and the sample size is typically smaller in organizational audits, the appropriate probability model for the mean is the Student’s t-distribution, not the normal (z) table (Casella & Berger, 2002). Use the t-distribution when σ is unknown and the sample is assumed to be drawn from a normally distributed population or when n is moderate; as n increases, the t-distribution approaches the standard normal (Agresti & Franklin, 2013).

Step 3 — Constructing the 90% and 99% confidence intervals in Excel

The two equivalent ways to construct a t-based CI in Excel are: (a) compute the t critical value using T.INV.2T(1−confidence, df) where df=n−1, then compute margin of error ME = t* × (s/√n) and CI = x̄ ± ME; or (b) use the CONFIDENCE.T(alpha, s, n) to obtain the margin of error directly (Microsoft Support, 2020a; 2020b). For a 90% CI, alpha = 0.10 and the t critical value is T.INV.2T(0.10, n−1); for a 99% CI, alpha = 0.01 and use T.INV.2T(0.01, n−1). Explicitly:

  • t90 = T.INV.2T(0.10, n−1)
  • ME90 = CONFIDENCE.T(0.10, s, n) or ME90 = t90 * s / SQRT(n)
  • CI90 = [x̄ − ME90, x̄ + ME90]
  • Repeat with alpha = 0.01 to obtain CI99.

Step 4 — Interval width and numerical interpretation

The width of an interval is computed as upper − lower = 2 × ME. Compare widths: CI99 will always be wider than CI90 for the same sample because a higher confidence level requires a larger critical value (t) and thus a larger ME (Agresti & Franklin, 2013; Casella & Berger, 2002). This relationship is quantitative: width99 = (t99/t90) × width90 (since s and n are identical across comparisons).

Decision rule about the 7-day promise

To decide whether the company should change the promised 7 days, examine whether the CI contains the value 7. If the entire confidence interval lies above 7 (lower limit > 7), there is evidence the true mean exceeds 7 and the promise should be reconsidered. If the interval lies entirely below 7 (upper limit

Practical recommendations and margin-of-error reduction strategies

If the 99% CI produces an unacceptably large ME, the team has several options to reduce the margin of error (Cochran, 1977; Hulley et al., 2013):

  • Increase sample size (n): ME decreases with √n. Doubling n reduces ME by factor 1/√2; to halve ME requires quadrupling n. This is the most direct and usually feasible option.
  • Reduce variance (s^2): Improve measurement precision or restrict sample heterogeneity (for example, stratify by route or shipping method). Lower s reduces ME proportionally.
  • Lower the confidence level: moving from 99% to 95% or 90% reduces t and thus ME, but at the cost of lower confidence.
  • Use paired or repeated measures or more efficient designs: when feasible, within-subject designs can reduce variability.

Excel implementation notes and verification

Implement the formulas in a dedicated worksheet so that all results are formula-driven (no hard-coded interim values). Typical cell formulas: B3 =AVERAGE(Data!A2:A101), B4 =STDEV.S(Data!A2:A101), B5 =COUNT(Data!A2:A101); B7 =T.INV.2T(0.10,B5-1); B8 =CONFIDENCE.T(0.10,B4,B5); B9 =B3+B8; B10 =B3-B8; similar for 99% with alpha=0.01. Verify calculations by recomputing margin of error via t*s/SQRT(n) and confirming identical results (Microsoft Support, 2020a; 2020b).

Conclusion

Following the steps above, the team will obtain both 90% and 99% confidence intervals for the population mean delivery time. The 99% interval will be wider. The practical recommendation about keeping or changing the 7-day promise depends strictly on whether 7 is inside or outside the interval bounds. If the lower bound exceeds 7, the company should adjust the promised delivery time; if the interval contains 7, the data do not provide sufficient evidence to change the promise at that confidence level. If the 99% ME is too large, increasing sample size and reducing variance are the preferred corrective actions to obtain a more precise estimate (Cochran, 1977; Hulley et al., 2013).

References

  • Agresti, A., & Franklin, C. (2013). Statistics: The Art and Science of Learning from Data. Pearson Education.
  • Altman, D. G. (1991). Practical Statistics for Medical Research. Chapman & Hall.
  • Cochran, W. G. (1977). Sampling Techniques (3rd ed.). John Wiley & Sons.
  • Casella, G., & Berger, R. L. (2002). Statistical Inference (2nd ed.). Duxbury.
  • Hulley, S. B., Cummings, S. R., Browner, W. S., Grady, D. G., & Newman, T. B. (2013). Designing Clinical Research (4th ed.). Lippincott Williams & Wilkins.
  • Moore, D. S., McCabe, G. P., & Craig, B. A. (2014). Introduction to the Practice of Statistics (8th ed.). W.H. Freeman.
  • Montgomery, D. C., & Runger, G. C. (2014). Applied Statistics and Probability for Engineers (6th ed.). Wiley.
  • Microsoft Support. (2020a). T.INV.2T function. https://support.microsoft.com/
  • Microsoft Support. (2020b). CONFIDENCE.T function. https://support.microsoft.com/
  • NIST/SEMATECH. (2012). e-Handbook of Statistical Methods. National Institute of Standards and Technology. https://www.itl.nist.gov/div898/handbook/

Related Assignments