Suppose we were interested in whether the current year's enrolments in the Languages faculty differed to previous years in terms of the distribution of students across different areas.  We usually expect a ratio of 4:2:2:1 across majors in Chinese, Arabic, Spanish and Indonesian respectively.  
For a sample of 180 students, the observed number of students enrolled in each major were: Chinese 72, Arabic 42, Spanish 37 and Indonesian 29.

How many were expected to major in Arabic?

**Choose the correct value.**

Question 8 options:

80

45

42

40

20

To calculate the expected number of students majoring in Arabic, given the usual expected ratio of 4:2:2:1 across the four majors (Chinese, Arabic, Spanish, and Indonesian), follow these steps:

1. **Determine the total ratio**: The ratio given is 4:2:2:1, so the total sum of the parts of the ratio is \( 4 + 2 + 2 + 1 = 9 \).

2. **Find the expected proportion for Arabic**: Arabic corresponds to 2 parts out of the total 9 parts.

3. **Calculate the expected number of students majoring in Arabic**:
   \[
   \text{Expected number of Arabic majors} = \frac{2}{9} \times 180 = \frac{360}{9} = 40
   \]

Thus, the expected number of students majoring in Arabic is **40**.

The correct option is **40**.

Let me know if you would like more details or have any questions.

Here are some related questions to expand your understanding:

1. How do you calculate the expected number of students for each major in this scenario?
2. What is a chi-square test, and how could it be applied to this problem?
3. How would the expected values change if the total sample size was different?
4. What assumptions must be met to apply the expected ratio in this context?
5. How could we interpret a large discrepancy between observed and expected values?

**Tip:** When dealing with expected ratios, ensure that you sum the ratio parts correctly before calculating expected values.

Learn how to calculate the expected number of students majoring in Arabic based on a given ratio and a sample size. This problem involves understanding ratios and applying them to determine expected values in statistical analysis. Suitable for students in Grades 9-12.

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