APSRTC/TSRTC: Passenger Satisfaction Survey

Solution

Conceptually, the required sample size for the APSRTC/TSRTC passenger satisfaction survey (covering routes like Hyderabad-Vijayawada, Tirupati-Hyderabad, Visakhapatnam-Hyderabad, and services to towns like Warangal, Kakinada, and Kurnool) is determined by the following key factors:

• 1. Desired Confidence Level:
  • This is the level of certainty that the sample estimate (proportion of satisfied travelers) accurately reflects the true proportion in the entire passenger population.
  • A higher confidence level (e.g., 95% or 99%) means we want to be more certain. To achieve higher certainty, a larger sample size is required. The problem specifies 95% confidence.
• 2. Desired Margin of Error (Precision):
  • This is how close we want our sample estimate to be to the true population proportion. It's the "+/-" value.
  • A smaller margin of error (e.g., +/- 3% as specified, versus +/- 5%) means we want a more precise estimate. To achieve higher precision (a smaller margin of error), a larger sample size is required.
• 3. Estimated Proportion of Satisfied Travelers (Population Proportion, p):
  • This is an estimate of the characteristic we are trying to measure (passenger satisfaction).
  • The sample size needed is largest when this proportion is assumed to be 50% (p=0.5). This is because p*(1-p) is maximized at p=0.5, leading to the highest variability.
  • If we have a preliminary estimate (e.g., from app reviews suggesting 60% satisfaction), using this estimate (p=0.6) can lead to a slightly smaller required sample size compared to assuming p=0.5. If the true proportion is closer to 0% or 100%, variability is lower, and a smaller sample is needed. Since the satisfaction is likely not at these extremes, considering a value like 0.5 or 0.6 is reasonable.
• 4. Population Size (N) (Less critical for large populations):
  • This is the total number of passengers using the Super Luxury and Garuda Plus services during the survey period across all Telugu states.
  • If the population is very large (as is likely for APSRTC/TSRTC passengers on major routes, especially during festival seasons like Sankranti), the sample size formula often uses an assumption of an infinite population, or the population size has a minimal impact on the required sample size.
  • A finite population correction factor can be applied if the calculated sample size is a significant fraction (e.g., >5%) of the total population, which would slightly reduce the required sample size. However, for a large and diverse passenger base across multiple cities and towns, this is often not a primary driver unless targeting very specific, small sub-groups.

In summary, to determine how many passengers to survey, we primarily need to decide on our desired confidence (95%), precision (+/- 3%), and make an educated guess or use a conservative estimate (like 50% or the preliminary 60%) for the expected satisfaction rate. These factors are then plugged into a standard sample size formula for proportions.

Solution

Having preliminary data suggesting approximately 60% satisfaction with the new APSRTC/TSRTC Super Luxury and Garuda Plus services (popular during Sankranti for travel to native places) affects the sample size calculation by potentially reducing the required sample size compared to using the most conservative assumption of 50% satisfaction.

The standard formula for sample size (n) for estimating a proportion (when the population is large) is:
n = (Z² * p * (1-p)) / E²
Where:

• Z = Z-score corresponding to the desired confidence level (e.g., 1.96 for 95% confidence).
• p = estimated population proportion (in this case, satisfaction rate).
• E = desired margin of error (e.g., 0.03 for +/- 3%).

The key term here that is affected by the estimate of satisfaction is p * (1-p), which represents the variance of a binomial distribution.

• Scenario 1: No Prior Estimate (Assume p = 0.5 or 50%)
  • When there is no prior estimate for the proportion, it is standard practice to use p = 0.5. This is the most conservative assumption because it maximizes the value of p * (1-p).
  • If p = 0.5, then p * (1-p) = 0.5 * (1-0.5) = 0.5 * 0.5 = 0.25.
  • This value maximizes the numerator in the sample size formula, thus yielding the largest (most conservative) required sample size to achieve the desired confidence and precision for the survey covering routes like Hyderabad-Vijayawada or Tirupati-Hyderabad.
• Scenario 2: Preliminary Estimate of p = 0.6 or 60% Satisfaction
  • If preliminary data (e.g., from app reviews for passengers traveling to Warangal, Kakinada, Kurnool, or during Sankranti) suggests p = 0.6.
  • Then, p * (1-p) = 0.6 * (1-0.6) = 0.6 * 0.4 = 0.24.

Effect on Sample Size Calculation:

Since 0.24 (from p=0.6) is less than 0.25 (from p=0.5), using the preliminary estimate of 60% satisfaction will result in a slightly smaller required sample size compared to assuming 50% satisfaction, all other factors (confidence level Z and margin of error E) remaining the same.

For example, with 95% confidence (Z=1.96) and +/-3% margin of error (E=0.03):

• Assuming p=0.5: n = (1.96² * 0.5 * 0.5) / 0.03² = (3.8416 * 0.25) / 0.0009 = 0.9604 / 0.0009 ≈ 1068 passengers.
• Assuming p=0.6: n = (1.96² * 0.6 * 0.4) / 0.03² = (3.8416 * 0.24) / 0.0009 = 0.921984 / 0.0009 ≈ 1025 passengers.

In this case, using the 60% estimate reduces the required sample size by approximately 43 passengers. While not a massive reduction, it does make the survey slightly more efficient. The further the estimated proportion (p) is from 0.5 (towards either 0 or 1), the smaller the p*(1-p) term becomes, and thus the smaller the required sample size.

Therefore, having a reasonable preliminary estimate of satisfaction (like 60%) allows for a more optimized (slightly smaller) sample size for the APSRTC/TSRTC survey, which can save time and resources.

Solution

There are significant practical tradeoffs between desiring higher precision (a smaller margin of error) or higher confidence, and the associated costs and feasibility of collecting larger samples for the APSRTC/TSRTC passenger satisfaction survey. These tradeoffs are particularly relevant when considering diverse passenger segments (IT professionals, students, families visiting temples like Tirumala) and routes (e.g., Hyderabad-Vijayawada, Tirupati-Hyderabad, Visakhapatnam-Hyderabad, and to smaller towns like Warangal, Kakinada, Kurnool).

Tradeoffs:

• 1. Higher Precision (Smaller Margin of Error, e.g., +/- 1% instead of +/- 3%):
  • Benefit: Provides a more exact estimate of the true proportion of satisfied passengers. This can lead to more fine-tuned decision-making.
  • Tradeoff (Cost/Feasibility):
    • Significantly Larger Sample Size Needed: The required sample size increases quadratically as the margin of error decreases (since E is squared in the denominator of the sample size formula). For instance, reducing the margin of error by half (e.g., from 4% to 2%) roughly quadruples the required sample size. Going from +/-3% to +/-1% would require approximately 9 times more passengers to be surveyed.
    • Increased Costs: More surveys mean higher costs for printing, distribution, data collection (e.g., interviewer time if not self-administered via app), data entry, and processing.
    • Increased Time: Collecting a much larger sample will take longer, potentially delaying insights needed for timely decisions (e.g., service changes before a peak season like Sankranti).
    • Logistical Complexity: Managing and ensuring quality for a very large survey across diverse routes and passenger types (IT professionals on weekends, students, families visiting Tirumala) becomes more challenging. Reaching specific segments in smaller towns like Warangal or Kurnool might require more effort per respondent.
• 2. Higher Confidence Level (e.g., 99% instead of 95%):
  • Benefit: Provides greater certainty that the true population proportion falls within the calculated confidence interval. Reduces the risk of the true value being outside the estimated range.
  • Tradeoff (Cost/Feasibility):
    • Larger Sample Size Needed: A higher confidence level requires a larger Z-score (e.g., Z=2.576 for 99% vs. Z=1.96 for 95%), which increases the required sample size.
    • Similar implications to higher precision regarding increased costs, time, and logistical complexity for surveying passengers on routes like Hyderabad-Vijayawada or to/from Tirupati.
• 3. Cost and Feasibility of Reaching Diverse Segments:
  • To ensure the results are representative of all passenger segments (IT professionals, students, families, travelers to/from smaller towns like Kakinada), the sampling plan needs to be robust.
  • Reaching a large enough sample within each specific segment to achieve high precision/confidence for that segment can be particularly costly and difficult. For example, getting a large, representative sample of weekend IT professional travelers on the Visakhapatnam-Hyderabad route might require surveying over many weekends.
  • The cost per completed survey might be higher for harder-to-reach segments or those in more remote locations connected by the bus network.
• 4. Practical Value of Incremental Precision/Confidence:
  • APSRTC/TSRTC management needs to consider if the business value of, for example, knowing satisfaction is 60% +/- 1% (very precise) versus 60% +/- 3% (less precise) justifies the significant additional cost and effort.
  • Often, a +/- 3% to +/- 5% margin of error at 95% confidence is deemed sufficient for many business decisions, providing a good balance between accuracy and practicality.

In conclusion, while higher precision and confidence are statistically desirable, they come at a direct cost of increased sample size, time, and complexity. For APSRTC/TSRTC, especially when surveying diverse passenger segments across many routes in Telangana and Andhra Pradesh (including those during busy Sankranti travel to native places or temple visits to Tirumala), a careful balance must be struck. The chosen levels (+/- 3% margin of error, 95% confidence) represent a common compromise, but if resources are highly constrained, management might even need to consider relaxing these slightly, fully understanding the impact on the reliability of the satisfaction estimates.