To investigate whether the observed employment status in the Sydney region follows the expected distribution given in the table, we can use a chi-square goodness-of-fit test. Here are the steps to find the test statistic:

1. **Define the hypotheses**:
   - Null Hypothesis \( H_0 \): The employment status distribution matches the expected distribution.
   - Alternative Hypothesis \( H_1 \): The employment status distribution does not match the expected distribution.

2. **Expected and Observed Values**:
   - We need the observed frequencies for each employment status to compute the chi-square statistic. Let me know if you have these values. Without observed values, I can only outline the process.

3. **Calculate the Chi-Square Test Statistic**:
   \[
   \chi^2 = \sum \frac{(O - E)^2}{E}
   \]
   where:
   - \( O \) is the observed frequency,
   - \( E \) is the expected frequency for each category (calculated as total count times each percentage from the table).

Could you provide the observed frequencies? With them, we can proceed to calculate the chi-square test statistic to two decimal places. 

Let me know if you'd like more details on the steps above or if you have questions on chi-square tests. 

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1. What are the types of statistical hypotheses in hypothesis testing?
2. How is the chi-square test used in categorical data analysis?
3. How does the chi-square distribution depend on degrees of freedom?
4. What assumptions underlie the chi-square goodness-of-fit test?
5. How would we interpret a significant chi-square result?

**Tip:** When interpreting the chi-square result, compare the test statistic to a critical value from the chi-square table based on the desired significance level and degrees of freedom.

Investigate whether the employment status distribution in Sydney matches the expected percentages: 30% Casual, 50% Full-time, 10% Part-time, and 10% Unemployed. Learn how to calculate the chi-square test statistic with a detailed solution, including hypothesis testing and statistical assumptions.

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