To determine the chi-square test statistic, [tex]\(\chi_0^2\)[/tex], we follow the formula:
[tex]\[
\chi_0^2 = \sum \frac{(O - E)^2}{E}
\][/tex]
where [tex]\(O\)[/tex] represents the observed frequencies and [tex]\(E\)[/tex] represents the expected frequencies. In this case, we have:
- Observed frequencies: [tex]\(O = [35, 21, 28, 28]\)[/tex]
- Expected frequencies: [tex]\(E = [28, 28, 28, 28]\)[/tex]
Let's calculate the chi-square test statistic step by step:
1. For Spring:
[tex]\[
\frac{(35 - 28)^2}{28} = \frac{7^2}{28} = \frac{49}{28} = 1.75
\][/tex]
2. For Summer:
[tex]\[
\frac{(21 - 28)^2}{28} = \frac{(-7)^2}{28} = \frac{49}{28} = 1.75
\][/tex]
3. For Fall:
[tex]\[
\frac{(28 - 28)^2}{28} = \frac{0^2}{28} = 0
\][/tex]
4. For Winter:
[tex]\[
\frac{(28 - 28)^2}{28} = \frac{0^2}{28} = 0
\][/tex]
Now, summing these values:
[tex]\[
1.75 + 1.75 + 0 + 0 = 3.5
\][/tex]
Thus, the chi-square test statistic is:
[tex]\[
\chi_0^2 = 3.5
\][/tex]
