To find the test statistic [tex]\(\chi^2\)[/tex] for the chi-square goodness-of-fit test, follow these steps:
1. Identify the observed (O) and expected (E) numbers of defects:
- For the range 0-100: [tex]\( O = 5, E = 5 \)[/tex]
- For the range 101-200: [tex]\( O = 7, E = 6 \)[/tex]
- For the range 201-300: [tex]\( O = 9, E = 7 \)[/tex]
- For the range 301-400: [tex]\( O = 11, E = 8 \)[/tex]
- For the range 401-500: [tex]\( O = 14, E = 10 \)[/tex]
2. Apply the chi-square formula to each observed and expected pair:
[tex]\[
\chi^2 = \sum \frac{(O - E)^2}{E}
\][/tex]
3. Calculate each term in the series:
- For the range 0-100:
[tex]\[
\frac{(O - E)^2}{E} = \frac{(5 - 5)^2}{5} = \frac{0^2}{5} = 0
\][/tex]
- For the range 101-200:
[tex]\[
\frac{(O - E)^2}{E} = \frac{(7 - 6)^2}{6} = \frac{1^2}{6} = \frac{1}{6} \approx 0.1667
\][/tex]
- For the range 201-300:
[tex]\[
\frac{(O - E)^2}{E} = \frac{(9 - 7)^2}{7} = \frac{2^2}{7} = \frac{4}{7} \approx 0.5714
\][/tex]
- For the range 301-400:
[tex]\[
\frac{(O - E)^2}{E} = \frac{(11 - 8)^2}{8} = \frac{3^2}{8} = \frac{9}{8} = 1.125
\][/tex]
- For the range 401-500:
[tex]\[
\frac{(O - E)^2}{E} = \frac{(14 - 10)^2}{10} = \frac{4^2}{10} = \frac{16}{10} = 1.6
\][/tex]
4. Sum the values you calculated:
[tex]\[
\chi^2 = 0 + 0.1667 + 0.5714 + 1.125 + 1.6 = 3.4631
\][/tex]
So, the test statistic [tex]\(\chi^2\)[/tex] for the chi-square goodness-of-fit test is approximately [tex]\(3.4631\)[/tex].
