Due to excessive pollution, GREEN Solutions Inc. is considering subsidizing public transportation to work for its employees. According to the manager…

To determine the probability of arriving on time (in 40 minutes or less), we calculate the Z-score for each transportation method, assuming a normal distribution. The Z-score measures how many standard deviations an element is from the mean.
1. Personal Vehicle: Mean (μ) = 39 min, Standard Deviation (σ) = 7 min, Target (X) = 40 min. Z = (X - μ) / σ = (40 - 39) / 7 ≈ 0.143. The probability P(Z ≤ 0.143) is approximately 55.7%.
2. Metro Public Transportation System (MPTS): Mean (μ) = 34 min, Standard Deviation (σ) = 21 min, Target (X) = 40 min. Z = (X - μ) / σ = (40 - 34) / 21 ≈ 0.286. The probability P(Z ≤ 0.286) is approximately 61.3%. Since 61.3% > 55.7%, the probability of arriving on time is higher using the MPTS.

A. The probability for personal vehicles (≈56%) is lower, not higher, than for the MPTS (≈61%).
C. The probabilities are demonstrably different (55.7% vs. 61.3%) and therefore not "about the same."
D. The problem is explicitly about the "average weekday commute," making weekend data irrelevant to the analysis.
E. This statement is a flawed generalization. While the MPTS has a higher standard deviation, its mean is significantly lower, which results in a higher probability of meeting the 40-minute target.

1. Montgomery, D. C. (2013). Introduction to statistical quality control (7th ed.). Wiley. Section 4.3, "The Normal Distribution," explains the use of the standard normal distribution and the Z-score transformation, Z = (X - μ) / σ, to compute probabilities for any normally distributed random variable (pp. 122-125).

2. The Pennsylvania State University. (n.d.). STAT 500: Applied Statistics, Lesson 4: Normal Distribution. Eberly College of Science. Retrieved from https://online.stat.psu.edu/stat500/lesson/4. This courseware details the process of "Standardization" (calculating a Z-score) to find the probability of a value falling within a certain range for any normal distribution.

3. Pyzdek, T., & Keller, P. A. (2018). The Six Sigma handbook (5th ed.). McGraw-Hill Education. Chapter 13, "Basic Control Charts," discusses how the normal distribution is fundamental to understanding process variation and capability, which involves calculating the probability of meeting specifications (pp. 265-267).