Parcel Shipping Efficiency

Solution & Explanation

To determine which parcel option is more reliable based on the probabilities of damage during shipment, we can utilize a two-proportion z-test. This statistical test is ideal for comparing the proportions of two independent groups, which in this scenario are the proportions of damaged parcels for types A and B.

Steps to Perform the Two-Proportion Z-Test

1. State the Hypotheses:

   • Null Hypothesis (H₀): The proportions of damaged packages are the same for both parcels, i.e., p = q.
   • Alternative Hypothesis (Hₐ): The proportions of damaged packages are different, i.e., p ≠ q.

2. Calculate the Sample Proportions:

   • For parcel A: $\hat{p}_A = \frac{40}{100} = 0.4$
   • For parcel B: $\hat{p}_B = \frac{60}{100} = 0.6$

3. Calculate the Pooled Proportion:

   • $\hat{p} = \frac{40 + 60}{200} = 0.5$

4. Calculate the Standard Error (SE):

   • $SE = \sqrt{\hat{p} \times (1 - \hat{p}) \times \left(\frac{1}{n_A} + \frac{1}{n_B}\right)}$
   • $SE = \sqrt{0.5 \times 0.5 \times \left(\frac{1}{100} + \frac{1}{100}\right)}$
   • $SE = \sqrt{0.25 \times 0.02} = \sqrt{0.005} \approx 0.0707$

5. Calculate the Z-Statistic:

   • $z = \frac{\hat{p}_A - \hat{p}_B}{SE}$
   • $z = \frac{0.4 - 0.6}{0.0707} \approx \frac{-0.2}{0.0707} \approx -2.83$

6. Determine the P-Value:

   • Using a standard normal distribution table or statistical software, find the p-value for z = -2.83.
   • The p-value is approximately 0.0047.

7. Make a Decision:

   • Compare the p-value to the significance level, typically $\alpha = 0.05$.
   • Since 0.0047 < 0.05, we reject the null hypothesis.

Conclusion

The statistical test indicates that there is a significant difference between the damage rates of parcels A and B. Specifically, parcel A, with a 40% damage rate, is statistically better than parcel B, which has a 60% damage rate. This suggests that parcel A is the more reliable option for shipping, assuming all other factors remain constant.