Krishna Waters: Bottling Process Quality Comparison

Solution

Comparing Process A (traditional Vijayawada process: 10,000 bottles, 100 defects) and Process B (new Warangal-developed technique: 500 bottles, 2 defects) for Krishna Waters presents several statistical challenges:

• 1. Highly Imbalanced Sample Sizes:
  • Process A has a sample size (n_A = 10,000) that is 20 times larger than Process B (n_B = 500).
  • Challenge: The estimate of the defect rate for Process A (p_A = 100/10,000 = 0.01) is quite precise due to the large sample. However, the estimate for Process B (p_B = 2/500 = 0.004) is based on a much smaller sample, making it inherently more uncertain and variable. A small change in the number of defects in Process B would lead to a proportionally larger change in its estimated defect rate.
• 2. Small Number of Defect Events in Process B:
  • Observing only 2 defects in Process B means we are dealing with a rare event within that smaller sample.
  • Challenge:
    • Unstable Rate Estimate: The estimated proportion (0.4%) is highly sensitive to small changes. If one more defect was found, the rate would jump to 0.6%; if one less, it would be 0.2%.
    • Violation of Assumptions for Standard Tests: Many common statistical tests for comparing proportions (like the chi-squared test or z-test for two proportions) rely on approximations that are valid only when the number of observed or expected events (and non-events) in each group is sufficiently large (e.g., typically >5). With only 2 defects in Process B, these approximations may not hold, leading to inaccurate p-values and potentially incorrect conclusions.
• 3. Difficulty in Assessing True Variability and Confidence:
  • Challenge: With very few defects, it's harder to get a reliable estimate of the true underlying defect probability for Process B and construct precise confidence intervals. The confidence interval for p_B will likely be very wide.
• 4. Potential for Low Statistical Power:
  • Challenge: Even if Process B is genuinely better, the small sample size and few observed defects might mean the test lacks sufficient statistical power to detect a statistically significant difference from Process A. Krishna Waters might incorrectly conclude there's no improvement when one actually exists (a Type II error).
• 5. Impact on Decision-Making:
  • Challenge: Making a decision to overhaul bottling processes across all plants in the Telugu states based on potentially unreliable statistical results is risky. An incorrect decision could lead to:
    • Unnecessary investment if Process B isn't truly superior.
    • Missed opportunity if a genuinely better process developed by Warangal engineers is discarded due to inconclusive small-sample results.

These challenges necessitate the use of statistical methods that are robust to small sample sizes and rare event counts to ensure a fair and reliable comparison between the two bottling processes.

Solution

For comparing the defect rates of the two bottling processes at Krishna Waters, given the data (Process A: 100 defects/10,000 bottles; Process B: 2 defects/500 bottles), the most appropriate statistical test would be Fisher's Exact Test.

Why a Standard Z-test for Two Proportions Might Be Problematic:

A standard z-test for two proportions relies on the normal approximation to the binomial distribution. This approximation is generally considered valid if certain conditions are met, typically:

• n₁p₁ ≥ 5, n₁(1-p₁) ≥ 5
• n₂p₂ ≥ 5, n₂(1-p₂) ≥ 5

(Where n is the sample size and p is the sample proportion for each group. Some statisticians use a threshold of 10.)

Let's check for Krishna Waters' data:

• For Process A (traditional Vijayawada process):
  • n_A = 10,000, Defects_A = 100, p_A = 0.01
  • n_Ap_A = 10000 * 0.01 = 100 (≥ 5)
  • n_A(1-p_A) = 10000 * 0.99 = 9900 (≥ 5)
  • The conditions are met for Process A.
• For Process B (new Warangal-developed technique):
  • n_B = 500, Defects_B = 2, p_B = 0.004
  • n_Bp_B = 500 * 0.004 = 2 (This is < 5)
  • n_B(1-p_B) = 500 * 0.996 = 498 (≥ 5)

Since the number of observed defects (n_Bp_B = 2) in Process B is less than the commonly accepted threshold of 5, the normal approximation used by the z-test may not be accurate for this group. Using a z-test in this scenario could lead to:

• An inaccurate p-value.
• Potentially misleading conclusions about the statistical significance of the difference.

A chi-squared test for independence (which is mathematically related to the z-test for two proportions) would also face similar issues because the expected count for the "Process B, Defect" cell under the null hypothesis (assuming equal proportions) would be (102/10500) * 500 ≈ 4.85, which is borderline or below the typical threshold of 5 for all expected cell counts.

Why Fisher's Exact Test is Most Appropriate:

• Handles Small Expected Frequencies: Fisher's Exact Test is an "exact" test, meaning it calculates the p-value based on the exact hypergeometric distribution of the data in a 2x2 contingency table, given the marginal totals. It does not rely on large-sample approximations.
• Designed for Rare Events: It is particularly well-suited for situations where one or more cells in the contingency table have small counts, as is the case with the 2 defects in Process B.
• Provides Reliable P-values: It gives a more accurate p-value when the assumptions for asymptotic tests (like z-test or chi-squared) are violated due to small cell counts.
• Application: We would set up a 2x2 table:

  	Defect	No Defect	Total
  Process A	100	9900	10000
  Process B	2	498	500
  Total	102	10398	10500

  Fisher's Exact Test would then determine the probability of observing this table, or one more extreme, if the defect rates were truly the same.

While Fisher's Exact Test can be conservative (less likely to find a significant result), its accuracy with small counts makes it the preferred choice here for Krishna Waters to make a statistically sound decision about their bottling processes.

Solution

To determine the minimum sample size required for Process B (the new Warangal-developed technique) to make a statistically valid conclusion for Krishna Waters, we need to perform a formal sample size calculation for comparing two proportions. This involves several key inputs:

• 1. Baseline Proportion (p₁ from Process A): The defect rate of the traditional Vijayawada process.
  p₁ = 100 defects / 10,000 bottles = 0.01 (or 1%).
• 2. Expected Proportion (p₂ for Process B) or Minimum Detectable Effect (MDE): This is the defect rate for Process B that Krishna Waters would consider a practically significant improvement.
  The pilot showed p_{B_observed} = 2/500 = 0.004. Management needs to decide what level of reduction they want to be able to detect. For example:
  • If they want to detect if p₂ is 0.005 (a 50% reduction from 0.01).
  • Or perhaps a more ambitious p₂ = 0.003.
  The smaller the difference (p₁ - p₂) they want to detect, the larger the sample size required. Let's assume they want to reliably detect if Process B can achieve a defect rate of 0.005 (0.5%).
• 3. Significance Level (α): The probability of making a Type I error (rejecting the null hypothesis when it is true; i.e., concluding Process B is better when it's not). This is typically set at 0.05. Since Krishna Waters is interested in whether Process B has fewer defects, a one-sided test is appropriate.
• 4. Statistical Power (1-β): The probability of making a correct decision when the alternative hypothesis is true (i.e., correctly concluding Process B is better if it truly has the target defect rate p₂). This is typically set at 0.80 (80%) or 0.90 (90%). Higher power means a lower chance of a Type II error (failing to detect a real improvement). Let's use 80% power.
• 5. Ratio of Sample Sizes (if applicable): If Process A's data is considered fixed from a very large historical dataset, we are essentially calculating n₂ (for Process B). If both are new samples, the ratio (e.g., 1:1) would be considered. Given the large n₁=10,000, we can often treat p₁ as well-estimated.

Calculation Approach:

Using statistical software (like R, Python's statsmodels, G*Power) or specialized online calculators for sample size for two independent proportions (one-sided test):

Inputs:

• p₁ = 0.01
• p₂ = 0.005 (the defect rate we want to be able to detect for Process B)
• α = 0.05 (one-sided)
• Power (1-β) = 0.80

A common formula for sample size per group (assuming equal sizes, which we can adjust for or use as a starting point for n₂ when n₁ is large) is:

n = [ (Z_α√(p_pooled(1-p_pooled)(1+1/k)) + Z_β√(p₁(1-p₁) + p₂(1-p₂)/k) ) / (p₁-p₂) ]²

Where p_pooled is the pooled proportion, k is the ratio n₁/n₂. For a simpler approximation when n₁ is very large, we can focus on the power to detect a difference from a known p₁.

Alternatively, using a calculator for comparing two proportions (one-sided):

• Proportion 1 (p1): 0.01
• Proportion 2 (p2): 0.005
• Alpha (one-sided): 0.05
• Power: 0.80
• Sample size ratio (n2/n1): This can be tricky. If we assume n1 is very large and fixed, we are essentially calculating n2 needed. If we were to run a new experiment with equal samples, the numbers would be different. For this context, we're focused on n2.

Using a standard online calculator (e.g., for a one-sided test of p1=0.01 vs p2=0.005, alpha=0.05, power=0.8, assuming n1 is large or we are looking for sample size n per group in a 1:1 scenario for simplicity of calculation here):

The required sample size for Process B (n_B) to detect this difference (from 1% down to 0.5%) with 80% power and 5% significance (one-sided) would typically be in the range of 2,500 to 3,000 bottles. (The exact number depends on the specific formula used by the calculator, whether continuity correction is applied, etc. For instance, some calculators might suggest around 2,755 per group for a 1:1 design. If n1 is very large, n2 would be in a similar ballpark.)

Recommendation:

To make a statistically valid conclusion for Krishna Waters about implementing the new Warangal-developed bottling technique across all plants in the Telugu states, I would recommend testing a minimum of approximately 2,800 bottles using Process B.

Rationale for this recommendation:

• This sample size would provide approximately 80% power to detect a decrease in the defect rate from the current 1% (Process A in Vijayawada) down to 0.5% (a 50% relative reduction) for Process B, using a one-sided test at a 5% significance level.
• A 0.5% defect rate (or a reduction of 0.005 absolute) is likely a practically significant improvement for a high-volume business like mineral water bottling, impacting costs and quality.
• While the initial 500 bottles showed an even lower rate (0.4%), a larger sample is needed to confirm this with confidence and ensure the result isn't due to random chance in a small sample.
• This balances the need for statistical confidence with the costs and time associated with extended testing. The company can adjust this based on their risk appetite and the precise MDE they deem critical. If they want higher power (e.g., 90%) or to detect an even smaller difference, the sample size would need to be larger.

It's also advisable to perform interim analyses if the testing is staged, though this requires more complex statistical planning (e.g., using alpha-spending functions) to avoid inflating the Type I error rate.