To perform a chi-square goodness-of-fit test, we need to calculate the test statistic [tex]\(\chi_0^2\)[/tex] using the formula:
[tex]\[
\chi_0^2 = \sum_k \frac{(O_k - E_k)^2}{E_k}
\][/tex]
where [tex]\(O_k\)[/tex] represents the observed frequencies and [tex]\(E_k\)[/tex] represents the expected frequencies for each category [tex]\(k\)[/tex].
Given the problem, we have:
- Expected frequencies ([tex]\(E_k\)[/tex]): 7 for each category (A, B, C, D)
- Observed frequencies ([tex]\(O_k\)[/tex]): 5, 7, 9, and 7 for categories A, B, C, and D respectively
Now, let's calculate the chi-square test statistic step-by-step:
1. Category A:
[tex]\[
\frac{(O_A - E_A)^2}{E_A} = \frac{(5 - 7)^2}{7} = \frac{(-2)^2}{7} = \frac{4}{7} \approx 0.571
\][/tex]
2. Category B:
[tex]\[
\frac{(O_B - E_B)^2}{E_B} = \frac{(7 - 7)^2}{7} = \frac{0^2}{7} = 0
\][/tex]
3. Category C:
[tex]\[
\frac{(O_C - E_C)^2}{E_C} = \frac{(9 - 7)^2}{7} = \frac{2^2}{7} = \frac{4}{7} \approx 0.571
\][/tex]
4. Category D:
[tex]\[
\frac{(O_D - E_D)^2}{E_D} = \frac{(7 - 7)^2}{7} = \frac{0^2}{7} = 0
\][/tex]
Now, sum these values to find the test statistic:
[tex]\[
\chi_0^2 = 0.571 + 0 + 0.571 + 0 = 1.142
\][/tex]
Rounding the result to three decimal places:
[tex]\[
\chi_0^2 \approx 1.143
\][/tex]
Thus, the chi-square test statistic is:
[tex]\[
\boxed{1.143}
\][/tex]
