To solve this question, let's complete the table step-by-step.

Step 1: Calculating $A$ (Observed Count for the age group 31-55)

We are given that the total observed count is 105. We already have observed counts for the other three age groups:

• $31$ for age group "less than 20"
• $29$ for age group "21-30"
• $22$ for age group "more than 55"

Using the total count, we can calculate $A$ as follows: $A = 105 - (31 + 29 + 22) = 105 - 82 = 23$

So, $A = 23$.

Step 2: Calculating $B$ (Expected Count for the age group more than 55)

The expected count for each age group is based on the percentages provided:

• Less than 20: 10% of 105
• 21-30: 10% of 105
• 31-55: 40% of 105
• More than 55: 40% of 105

For age group "more than 55": $B = 0.40 \times 105 = 42$

So, $B = 42$.

Step 3: Calculating $C$ (chi-squared value for age group less than 20)

The chi-squared statistic for each group is calculated by the formula: $\chi^2 = \frac{(\text{Observed Count} - \text{Expected Count})^2}{\text{Expected Count}}$

For age group "less than 20": $C = \frac{(31 - 10.5)^2}{10.5} = \frac{(20.5)^2}{10.5} = \frac{420.25}{10.5} = 40.0238$

So, $C \approx 40.024$ (rounded to three decimal places).

Completed Table

Now we can fill in the missing values in the table:

| | 1 (less than 20) | 2 (21-30) | 3 (31-55) | 4 (more than 55