Question 1199632
### Part a) Probability that Ziteck keeps the shipment if the defect rate is 5%
This is a binomial distribution problem where:

- \( n = 10 \) (sample size),
- \( p = 0.05 \) (probability of a defective part),
- \( X \) is the number of defective parts in the sample.

Ziteck keeps the shipment if \( X \leq 2 \). The probability of this is:

\[
P(X \leq 2) = P(X = 0) + P(X = 1) + P(X = 2)
\]

The probability mass function of a binomial distribution is:

\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]

So:

1. \( P(X = 0) = \binom{10}{0} (0.05)^0 (0.95)^{10} \)
2. \( P(X = 1) = \binom{10}{1} (0.05)^1 (0.95)^9 \)
3. \( P(X = 2) = \binom{10}{2} (0.05)^2 (0.95)^8 \)

Summing these gives \( P(X \leq 2) \). Let's calculate.

The probability that Ziteck will keep the shipment if the defect rate is 5% is approximately **0.9885** (98.85%).

---

### Part b) Probability that Ziteck keeps the shipment if the defect rate is 10%
If the defect rate is \( p = 0.10 \), we use the same approach, calculating \( P(X \leq 2) \) for \( n = 10 \) and \( p = 0.10 \).

The probability that Ziteck will keep the shipment if the defect rate is 10% is approximately **0.9298** (92.98%).

---

### Part c) Commentary on the sampling plan

1. **Sample size**: A sample size of 10 is relatively small. While it simplifies the process of inspecting shipments, it may not provide a reliable estimate of the overall defect rate in large shipments, especially if the defect rate is close to the threshold.

2. **Acceptance/rejection point**: The decision rule (\( X \leq 2 \)) makes it highly likely that Ziteck will accept shipments, even if the defect rate is higher than 5%. For example, with a 10% defect rate, there is still a 92.98% chance of accepting the shipment.

3. **Favors supplier or Ziteck?**:
   - The plan appears to favor the supplier, as it allows for a high likelihood of shipment acceptance even when the defect rate exceeds the contractual limit.
   - From Ziteck's perspective, this increases the risk of receiving shipments with a defect rate that exceeds the acceptable threshold, potentially leading to quality issues in their operations.

**Recommendation**: Ziteck could consider increasing the sample size or lowering the acceptance threshold (\( X \leq 1 \)) to reduce the risk of accepting defective shipments. These changes would provide a better balance between supplier quality assurance and Ziteck's operational needs.