To find the chi-square test statistic, [tex]\(\chi_0^2\)[/tex], for the given data, we use the chi-square goodness-of-fit formula:
[tex]\[
\chi_0^2 = \sum \frac{(O - E)^2}{E}
\][/tex]
where [tex]\(O\)[/tex] represents the observed frequency and [tex]\(E\)[/tex] represents the expected frequency for each category.
Given the data:
- Expected counts for each season (E): 27 (for Spring, Summer, Fall, and Winter)
- Observed counts for each season (O): 15 (Spring), 18 (Summer), 32 (Fall), 43 (Winter)
We will calculate the chi-square value for each season and then sum them up.
Step-by-Step Calculation:
1. Spring:
[tex]\[
\chi_{\text{Spring}}^2 = \frac{(15 - 27)^2}{27} = \frac{(-12)^2}{27} = \frac{144}{27} = 5.333
\][/tex]
2. Summer:
[tex]\[
\chi_{\text{Summer}}^2 = \frac{(18 - 27)^2}{27} = \frac{(-9)^2}{27} = \frac{81}{27} = 3.000
\][/tex]
3. Fall:
[tex]\[
\chi_{\text{Fall}}^2 = \frac{(32 - 27)^2}{27} = \frac{(5)^2}{27} = \frac{25}{27} \approx 0.926
\][/tex]
4. Winter:
[tex]\[
\chi_{\text{Winter}}^2 = \frac{(43 - 27)^2}{27} = \frac{(16)^2}{27} = \frac{256}{27} \approx 9.481
\][/tex]
Total chi-square statistic:
[tex]\[
\chi_0^2 = 5.333 + 3.000 + 0.926 + 9.481 = 18.740
\][/tex]
Round to three decimal places:
[tex]\[
\chi_0^2 = 18.741
\][/tex]
Therefore, the chi-square test statistic is:
[tex]\[
\chi_0^2 = 18.741
\][/tex]
