Sure, let's walk through the steps for solving this problem.
### Step-by-Step Solution:
#### (a) Degrees of Freedom Calculation:
To calculate the degrees of freedom for a chi-square test of independence, we use the formula:
[tex]\[ \text{Degrees of freedom} = (r - 1) \times (c - 1) \][/tex]
where [tex]\( r \)[/tex] is the number of rows and [tex]\( c \)[/tex] is the number of columns in the table.
From the given table:
1. [tex]\( r = 3 \)[/tex] (since there are 3 rows)
2. [tex]\( c = 3 \)[/tex] (since there are 3 columns)
Plugging in these values:
[tex]\[ \text{Degrees of freedom} = (3 - 1) \times (3 - 1) = 2 \times 2 = 4 \][/tex]
So, the degrees of freedom is:
[tex]\[ \text{Degrees of freedom} = 4 \][/tex]
#### (b) Calculating the p-value:
To find the p-value, we use the chi-square distribution with the chi-square statistic provided.
We're given:
- Chi-square statistic [tex]\( \chi^2 = 31.316 \)[/tex]
- Degrees of freedom [tex]\( \text{df} = 4 \)[/tex]
Looking up the p-value for the chi-square statistic with these degrees of freedom using chi-square distribution tables or a statistical software, we get:
[tex]\[ p\text{-value} \approx 2.639002855242545 \times 10^{-6} \][/tex]
Therefore, the p-value is:
[tex]\[ p\text{-value} \approx 2.639002855242545 \times 10^{-6} \][/tex]
### Final Answers:
(a) Degrees of freedom: [tex]\( \boxed{4} \)[/tex]
(b) p-value: [tex]\( \boxed{2.639002855242545 \times 10^{-6}} \)[/tex]
