To recommend a minimum sample size $n$ for Aaron and Aayush's survey, we need to consider the statistical method they'll likely use to investigate if the true proportion of QANTAS passengers who lost their bag has changed since pre-COVID. A common approach for such an investigation is hypothesis testing, where the goal is to test whether the current proportion is different from the pre-COVID proportion.

The formula for determining the minimum sample size required for a proportion hypothesis test (such as a two-proportion z-test) is:

$n = \left( \frac{Z_{\alpha/2} \cdot \sqrt{p(1-p)}}{E} \right)^2$

Where:

• $Z_{\alpha/2}$ is the critical value from the standard normal distribution for a given confidence level.
• $p$ is the assumed proportion (from the pre-COVID period, which is 0.005 based on the given data).
• $E$ is the margin of error (desired precision of the estimate).

Step 1: Identify key parameters

• Pre-COVID proportion $p = 0.005$ (5 passengers per 1,003 passengers, or $\frac{5}{1003}$).
• Confidence level: Let's assume a 95% confidence level, which is common in such studies. The corresponding $Z$-value for a 95% confidence level is $Z_{\alpha/2} \approx 1.96$.
• Margin of error $E$: This will depend on the precision Aaron and Aayush want. Let's assume they want a margin of error of $0.01$ (1%).

Step 2: Plug in the values

Using the formula for minimum sample size:

$n = \left( \frac{1.96 \cdot \sqrt{0.005 \cdot (1 - 0.005)}}{0.01} \right)^2$

Now, let's calculate the required minimum sample size $n$.The minimum sample size required for Aaron and Aayush's survey is 192 passengers. This sample size ensures that their estimate of the true proportion of QANTAS passengers who lost their baggage will have a margin of error of 1% with 95% confidence.

Would you like more details on how this result was derived, or do you have any other questions?

Here are some related questions you might consider:

1. What if Aaron and Aayush want a different margin of error, like 2%? How would that affect the sample size?
2. How does the choice of confidence level (e.g., 90% vs 95%) impact the sample size?
3. How do Aaron and Aayush ensure that their sample is representative of all QANTAS passengers?
4. What statistical tests could Aaron and Aayush use to compare their new sample proportion with the pre-COVID proportion?
5. How would changes in passenger volume due to COVID affect the interpretation of the results?
6. What assumptions are made when calculating the sample size, and how could those assumptions be checked?
7. What if the actual lost baggage rate is higher than expected? How should Aaron and Aayush adjust their analysis?
8. How does the variability in lost baggage rates across different airports or routes affect the study's conclusions?

Tip: Always verify that the sample is representative of the broader population to ensure accurate results in surveys and hypothesis tests.