A human resource manager is trying to reduce the percentage of employees resigning at the organisation's two factories in Brisbane and Christchurch. She implements an employee loyalty scheme at the Christchurch factory in which employees receive a financial bonus for staying at the factory for over five years. Over the next 12-month period, 12 out of 345 employees resign at the Christchurch factory, compared to 32 out of 890 employees in Brisbane. Calculate the value of the test statistic used to test whether the percentage of employees resigning in Christchurch is lower than that of Brisbane (round your final answer to one decimal place). Assume population 1 = Brisbane factory and population 2 = Christchurch factory. 

To calculate the test statistic for comparing the proportion of employees resigning between the two factories, we follow these steps:

### Step 1: Define the Proportions
- Let \( p_1 \) be the proportion of employees resigning in Brisbane.
- Let \( p_2 \) be the proportion of employees resigning in Christchurch.

Given:
- \( n_1 = 890 \) (number of employees in Brisbane)
- \( x_1 = 32 \) (number of resignations in Brisbane)
- \( n_2 = 345 \) (number of employees in Christchurch)
- \( x_2 = 12 \) (number of resignations in Christchurch)

The sample proportions are:
\[
\hat{p}_1 = \frac{x_1}{n_1} = \frac{32}{890} \approx 0.0360
\]
\[
\hat{p}_2 = \frac{x_2}{n_2} = \frac{12}{345} \approx 0.0348
\]

### Step 2: Calculate the Pooled Proportion
The pooled proportion \( \hat{p} \) is calculated as:
\[
\hat{p} = \frac{x_1 + x_2}{n_1 + n_2} = \frac{32 + 12}{890 + 345} = \frac{44}{1235} \approx 0.0356
\]

### Step 3: Calculate the Standard Error (SE)
The standard error of the difference between the two proportions is:
\[
SE = \sqrt{\hat{p} \left(1 - \hat{p}\right) \left(\frac{1}{n_1} + \frac{1}{n_2}\right)}
\]
Substituting the values:
\[
SE = \sqrt{0.0356 \times (1 - 0.0356) \times \left(\frac{1}{890} + \frac{1}{345}\right)} \approx 0.0133
\]

### Step 4: Calculate the Test Statistic (Z)
The test statistic \( Z \) is calculated as:
\[
Z = \frac{\hat{p}_1 - \hat{p}_2}{SE} = \frac{0.0360 - 0.0348}{0.0133} \approx \frac{0.0012}{0.0133} \approx 0.09
\]

### Final Answer
The value of the test statistic is approximately \( Z = 0.1 \) (rounded to one decimal place).

Would you like more details or have any questions?

Here are 5 related questions to expand on this information:
1. How would you interpret this test statistic in the context of a hypothesis test?
2. What are the assumptions required for the validity of this test statistic?
3. How does changing the sample size impact the test statistic?
4. What would be the effect of a different pooled proportion on the result?
5. How would the test statistic change if more employees resigned in either factory?

**Tip:** When interpreting the test statistic, always consider its magnitude in relation to the critical value, especially when making decisions based on hypothesis tests.

Calculate and interpret the test statistic to compare the employee resignation rates between Brisbane and Christchurch factories using statistical inference methods. Includes step-by-step calculation and interpretation of proportions and standard error.

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