Sure, let's calculate the chi-square statistic for this goodness-of-fit test step-by-step using the given observed and expected counts.
1. Given Data:
- Observed counts ([tex]\(O\)[/tex]):
- [tex]\(O_A = 29\)[/tex]
- [tex]\(O_B = 49\)[/tex]
- [tex]\(O_C = 45\)[/tex]
- Expected counts ([tex]\(E\)[/tex]):
- [tex]\(E_A = 36.9\)[/tex]
- [tex]\(E_B = 36.9\)[/tex]
- [tex]\(E_C = 49.2\)[/tex]
2. Formula for Chi-square Statistic:
The formula for the chi-square statistic is:
[tex]\[
\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}
\][/tex]
where [tex]\(O_i\)[/tex] is the observed count, [tex]\(E_i\)[/tex] is the expected count, and the sum is over all categories.
3. Calculate Individual Components:
- For category A:
[tex]\[
\chi^2_A = \frac{(O_A - E_A)^2}{E_A} = \frac{(29 - 36.9)^2}{36.9} \approx 1.6913
\][/tex]
- For category B:
[tex]\[
\chi^2_B = \frac{(O_B - E_B)^2}{E_B} = \frac{(49 - 36.9)^2}{36.9} \approx 3.9678
\][/tex]
- For category C:
[tex]\[
\chi^2_C = \frac{(O_C - E_C)^2}{E_C} = \frac{(45 - 49.2)^2}{49.2} \approx 0.3585
\][/tex]
4. Sum the Contributions:
[tex]\[
\chi^2 = \chi^2_A + \chi^2_B + \chi^2_C = 1.6913 + 3.9678 + 0.3585 \approx 6.0176
\][/tex]
5. Round the Result:
The chi-square statistic rounded to one decimal place is:
[tex]\[
\chi^2 \approx 6.0
\][/tex]
So, the chi-square statistic for this test is [tex]\(\chi^2 = 6.0\)[/tex].
