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Sagot :
To calculate the chi-square statistic for a chi-square test for independence, you can use the observed and expected frequencies provided in the table. Follow these steps:
1. Identify the Observed and Expected Values:
The observed values are:
[tex]\[ \begin{array}{ccc} 30 & 31 & 39 \\ 25 & 51 & 24 \\ 150 & 72 & 78 \\ \end{array} \][/tex]
The expected values are:
[tex]\[ \begin{array}{ccc} 41 & 30.8 & 28.2 \\ 41 & 30.8 & 28.2 \\ 123 & 92.4 & 84.6 \\ \end{array} \][/tex]
2. Apply the Chi-Square Formula:
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] are the observed frequencies and [tex]\( E_i \)[/tex] are the expected frequencies.
3. Calculate [tex]\((O_i - E_i)\)[/tex] and [tex]\(\frac{(O_i - E_i)^2}{E_i}\)[/tex]:
- For cell (1, A): [tex]\( O = 30 \)[/tex], [tex]\( E = 41 \)[/tex]
[tex]\[ (30 - 41)^2 / 41 = (-11)^2 / 41 = 121 / 41 = 2.951 \][/tex]
- For cell (1, B): [tex]\( O = 31 \)[/tex], [tex]\( E = 30.8 \)[/tex]
[tex]\[ (31 - 30.8)^2 / 30.8 = (0.2)^2 / 30.8 = 0.04 / 30.8 = 0.001 \][/tex]
- For cell (1, C): [tex]\( O = 39 \)[/tex], [tex]\( E = 28.2 \)[/tex]
[tex]\[ (39 - 28.2)^2 / 28.2 = (10.8)^2 / 28.2 = 116.64 / 28.2 = 4.137 \][/tex]
- For cell (2, A): [tex]\( O = 25 \)[/tex], [tex]\( E = 41 \)[/tex]
[tex]\[ (25 - 41)^2 / 41 = (-16)^2 / 41 = 256 / 41 = 6.244 \][/tex]
- For cell (2, B): [tex]\( O = 51 \)[/tex], [tex]\( E = 30.8 \)[/tex]
[tex]\[ (51 - 30.8)^2 / 30.8 = (20.2)^2 / 30.8 = 408.04 / 30.8 = 13.247 \][/tex]
- For cell (2, C): [tex]\( O = 24 \)[/tex], [tex]\( E = 28.2 \)[/tex]
[tex]\[ (24 - 28.2)^2 / 28.2 = (-4.2)^2 / 28.2 = 17.64 / 28.2 = 0.625 \][/tex]
- For cell (3, A): [tex]\( O = 150 \)[/tex], [tex]\( E = 123 \)[/tex]
[tex]\[ (150 - 123)^2 / 123 = (27)^2 / 123 = 729 / 123 = 5.927 \][/tex]
- For cell (3, B): [tex]\( O = 72 \)[/tex], [tex]\( E = 92.4 \)[/tex]
[tex]\[ (72 - 92.4)^2 / 92.4 = (-20.4)^2 / 92.4 = 416.16 / 92.4 = 4.504 \][/tex]
- For cell (3, C): [tex]\( O = 78 \)[/tex], [tex]\( E = 84.6 \)[/tex]
[tex]\[ (78 - 84.6)^2 / 84.6 = (-6.6)^2 / 84.6 = 43.56 / 84.6 = 0.515 \][/tex]
4. Sum All These Values:
[tex]\[ \chi^2 = 2.951 + 0.001 + 4.137 + 6.244 + 13.247 + 0.625 + 5.927 + 4.504 + 0.515 = 38.152 \][/tex]
Therefore, the chi-square statistic is:
[tex]\[ \chi^2 = 38.152 \][/tex]
1. Identify the Observed and Expected Values:
The observed values are:
[tex]\[ \begin{array}{ccc} 30 & 31 & 39 \\ 25 & 51 & 24 \\ 150 & 72 & 78 \\ \end{array} \][/tex]
The expected values are:
[tex]\[ \begin{array}{ccc} 41 & 30.8 & 28.2 \\ 41 & 30.8 & 28.2 \\ 123 & 92.4 & 84.6 \\ \end{array} \][/tex]
2. Apply the Chi-Square Formula:
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] are the observed frequencies and [tex]\( E_i \)[/tex] are the expected frequencies.
3. Calculate [tex]\((O_i - E_i)\)[/tex] and [tex]\(\frac{(O_i - E_i)^2}{E_i}\)[/tex]:
- For cell (1, A): [tex]\( O = 30 \)[/tex], [tex]\( E = 41 \)[/tex]
[tex]\[ (30 - 41)^2 / 41 = (-11)^2 / 41 = 121 / 41 = 2.951 \][/tex]
- For cell (1, B): [tex]\( O = 31 \)[/tex], [tex]\( E = 30.8 \)[/tex]
[tex]\[ (31 - 30.8)^2 / 30.8 = (0.2)^2 / 30.8 = 0.04 / 30.8 = 0.001 \][/tex]
- For cell (1, C): [tex]\( O = 39 \)[/tex], [tex]\( E = 28.2 \)[/tex]
[tex]\[ (39 - 28.2)^2 / 28.2 = (10.8)^2 / 28.2 = 116.64 / 28.2 = 4.137 \][/tex]
- For cell (2, A): [tex]\( O = 25 \)[/tex], [tex]\( E = 41 \)[/tex]
[tex]\[ (25 - 41)^2 / 41 = (-16)^2 / 41 = 256 / 41 = 6.244 \][/tex]
- For cell (2, B): [tex]\( O = 51 \)[/tex], [tex]\( E = 30.8 \)[/tex]
[tex]\[ (51 - 30.8)^2 / 30.8 = (20.2)^2 / 30.8 = 408.04 / 30.8 = 13.247 \][/tex]
- For cell (2, C): [tex]\( O = 24 \)[/tex], [tex]\( E = 28.2 \)[/tex]
[tex]\[ (24 - 28.2)^2 / 28.2 = (-4.2)^2 / 28.2 = 17.64 / 28.2 = 0.625 \][/tex]
- For cell (3, A): [tex]\( O = 150 \)[/tex], [tex]\( E = 123 \)[/tex]
[tex]\[ (150 - 123)^2 / 123 = (27)^2 / 123 = 729 / 123 = 5.927 \][/tex]
- For cell (3, B): [tex]\( O = 72 \)[/tex], [tex]\( E = 92.4 \)[/tex]
[tex]\[ (72 - 92.4)^2 / 92.4 = (-20.4)^2 / 92.4 = 416.16 / 92.4 = 4.504 \][/tex]
- For cell (3, C): [tex]\( O = 78 \)[/tex], [tex]\( E = 84.6 \)[/tex]
[tex]\[ (78 - 84.6)^2 / 84.6 = (-6.6)^2 / 84.6 = 43.56 / 84.6 = 0.515 \][/tex]
4. Sum All These Values:
[tex]\[ \chi^2 = 2.951 + 0.001 + 4.137 + 6.244 + 13.247 + 0.625 + 5.927 + 4.504 + 0.515 = 38.152 \][/tex]
Therefore, the chi-square statistic is:
[tex]\[ \chi^2 = 38.152 \][/tex]
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