Unfortunately passengers in Australia are experiencing issues with lost baggage with the major domestic airline QANTAS. Pre-COVID, the true proportion, Australia wide, of passengers who lost their bag with QANTAS was 5 passengers per 1,003 passengers, i.e.,   51003=5001003
% .

In April of 2022 the Data is Us market research firm was hired to investigate if the true proportion of all QANTAS passengers who lost their bag has changed since Pre-COVID. Also, Aaron and Aayush, the leaders of this project with Data is Us, want to provide an updated (from Pre-COVID) estimate for the true proportion of all QANTAS passengers who lost their bag.

Your job is to help Aaron and Aayush with the required calculations and explanations for this brief to present to the client, domestic airline QANTAS.

First, Aaron and Aayush decide to collect data: Aaron spends three days in airports in NSW while Aayush spends three days in airports in VIC (NSW and VIC are states/territories in Australia). During those days, they respectively ask passengers of QANTAS arriving into NSW and VIC if their baggage was lost.

(a) What would you recommend as the MINIMUM sample size  n
for their survey based on the Pre-COVID true proportion, given that Aaron and Aayush want to use their sample to investigate, using a method taught in MATH1041, if the true proportion of all QANTAS passengers who lost their bag had changed since Pre-COVID?

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.

Learn how to calculate the minimum sample size required for a proportion hypothesis test to investigate changes in the proportion of QANTAS passengers who lost their baggage since Pre-COVID. This problem involves statistical concepts such as hypothesis testing, sampling, and estimating proportions. Suitable for college-level students and researchers.

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