A random sample of 130 students enrolled in statistics units in recent semesters at Macquarie University has been selected to study use of practice quizzes and academic performance. Some of the variables recorded for each student are listed below.

Variable		Description
subid		Subject ID
intDom		domestic or international
numQuizzes		the number of practice quizzes attempted
unitMark		the final unit mark
ageGroupCat		the age of students

Part 1

Research question: We wish to investigate whether the age of students enrolled in statistics units in recent semesters at Macquarie University is distributed according to the table below.
the age of students | 1 (less than 19) | 2 (19-21) | 3 (22-24) | 4 (more than 24)
Percentage          |               10 |        40 |        30 |               20
Please complete the following table calculating the values of A, B and C.

               | 1 (less than 19) | 2 (19-21) | 3 (22-24) | 4 (more than 24) | Total 
Observed Count | A=***            | 37        | 36        | 23               | 130   
Expected Count | 13               | 52        | B=***     | 26               | 130   
chi squared    | 33.923           | 4.327     | 0.231     | C=***            | 38.827
A [integer]:	
B [use at least 3dp]:	

To complete this table, we need to calculate the missing values \( A \), \( B \), and \( C \). Here's the approach:

### Step 1: Determine Observed Count \( A \)
The total observed count for all categories is 130. Since counts for categories 2, 3, and 4 are given (37, 36, and 23 respectively), we can find \( A \) by subtracting these from the total:

\[
A = 130 - (37 + 36 + 23) = 130 - 96 = 34
\]

### Step 2: Determine Expected Count \( B \)
The expected counts are calculated based on the percentages given for each age group. For category 3 (22-24), the expected count \( B \) is based on 30% of the total sample of 130 students:

\[
B = 0.30 \times 130 = 39
\]

### Step 3: Determine Chi-Squared Value \( C \)
The chi-squared statistic for each category is calculated as:

\[
\chi^2 = \frac{(O - E)^2}{E}
\]

where \( O \) is the observed count, and \( E \) is the expected count. To find \( C \), which is the chi-squared value for the fourth category:

\[
C = \frac{(23 - 26)^2}{26} = \frac{9}{26} = 0.346
\]

### Summary of Values
1. **A (Observed Count for category 1)**: \( A = 34 \)
2. **B (Expected Count for category 3)**: \( B = 39.000 \)
3. **C (Chi-squared for category 4)**: \( C = 0.346 \)

Would you like more detailed explanations on chi-squared calculations? 

---

### Further Questions to Explore:
1. What does the chi-squared value represent in this context?
2. How does changing sample size affect the chi-squared value?
3. Why is the expected count derived from percentage distributions?
4. How would the result change if observed counts differed slightly?
5. What conclusion could be drawn from a high total chi-squared value?

#### Tip:
When calculating expected counts, always ensure they are proportional to the sample size and distribution percentages.

A random sample of 130 students enrolled in statistics units in recent semesters at Macquarie University has been selected to study the use of practice quizzes and academic performance. Variables include student ID, international or domestic status, number of quizzes attempted, final unit mark, and age group category. Part 1: Investigate whether student age is distributed as per the given table, calculating the values of A, B, and C in the chi-squared test table.

Calculate chi-squared values and assess age distribution among 130 statistics students at Macquarie University. This example explores observed and expected counts, using chi-squared calculations to verify the distribution. Key for understanding goodness-of-fit tests and expected value calculations.

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