Answer :
To determine whether favorite food depends on gender, we will perform a Chi-Square Test of Independence using the provided contingency table. We'll go through the following systematic steps:
1. Formulate the Hypotheses:
- Null hypothesis ([tex]\(H_0\)[/tex]): Gender and favorite food are independent.
- Alternative hypothesis ([tex]\(H_a\)[/tex]): Gender and favorite food are dependent.
2. Construct the Contingency Table:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline & \text{Sushi} & \text{Pizza} & \text{Hamburgers} & \text{Row Total} \\ \hline \text{Men} & 5 & 24 & 21 & 50 \\ \hline \text{Women} & 17 & 21 & 8 & 46 \\ \hline \text{Column Total} & 22 & 45 & 29 & 96 \\ \hline \end{array} \][/tex]
3. Compute the Expected Frequencies:
- The expected frequency for each cell is calculated using the formula:
[tex]\[ E_{ij} = \frac{(\text{Row Total for row } i) \times (\text{Column Total for column } j)}{\text{Grand Total}} \][/tex]
Calculating the expected frequencies for each cell:
- Expected frequency for Men who prefer Sushi ([tex]\(E_{11}\)[/tex]):
[tex]\[ E_{11} = \frac{50 \times 22}{96} \approx 11.5 \][/tex]
- Expected frequency for Men who prefer Pizza ([tex]\(E_{12}\)[/tex]):
[tex]\[ E_{12} = \frac{50 \times 45}{96} \approx 23.4 \][/tex]
- Expected frequency for Men who prefer Hamburgers ([tex]\(E_{13}\)[/tex]):
[tex]\[ E_{13} = \frac{50 \times 29}{96} \approx 15.1 \][/tex]
- Expected frequency for Women who prefer Sushi ([tex]\(E_{21}\)[/tex]):
[tex]\[ E_{21} = \frac{46 \times 22}{96} \approx 10.5 \][/tex]
- Expected frequency for Women who prefer Pizza ([tex]\(E_{22}\)[/tex]):
[tex]\[ E_{22} = \frac{46 \times 45}{96} \approx 21.6 \][/tex]
- Expected frequency for Women who prefer Hamburgers ([tex]\(E_{23}\)[/tex]):
[tex]\[ E_{23} = \frac{46 \times 29}{96} \approx 13.9 \][/tex]
4. Calculate the Chi-Square Test Statistic:
- The formula for the test statistic is:
[tex]\[ \chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}} \][/tex]
Where [tex]\(O_{ij}\)[/tex] is the observed frequency and [tex]\(E_{ij}\)[/tex] is the expected frequency.
Summing these values:
- Contribution from Men who prefer Sushi:
[tex]\[ \frac{(5 - 11.5)^2}{11.5} \approx 3.67 \][/tex]
- Contribution from Men who prefer Pizza:
[tex]\[ \frac{(24 - 23.4)^2}{23.4} \approx 0.02 \][/tex]
- Contribution from Men who prefer Hamburgers:
[tex]\[ \frac{(21 - 15.1)^2}{15.1} \approx 2.31 \][/tex]
- Contribution from Women who prefer Sushi:
[tex]\[ \frac{(17 - 10.5)^2}{10.5} \approx 4.01 \][/tex]
- Contribution from Women who prefer Pizza:
[tex]\[ \frac{(21 - 21.6)^2}{21.6} \approx 0.02 \][/tex]
- Contribution from Women who prefer Hamburgers:
[tex]\[ \frac{(8 - 13.9)^2}{13.9} \approx 2.35 \][/tex]
Adding these contributions gives us:
[tex]\[ \chi^2 \approx 12.4 \][/tex]
5. State the Conclusion:
- The Chi-Square test statistic [tex]\(\chi^2\)[/tex] is approximately 12.4.
Therefore, the computed test statistic, rounded to one decimal place, is [tex]\(\chi^2 = 12.4\)[/tex].
1. Formulate the Hypotheses:
- Null hypothesis ([tex]\(H_0\)[/tex]): Gender and favorite food are independent.
- Alternative hypothesis ([tex]\(H_a\)[/tex]): Gender and favorite food are dependent.
2. Construct the Contingency Table:
[tex]\[ \begin{array}{|c|c|c|c|c|} \hline & \text{Sushi} & \text{Pizza} & \text{Hamburgers} & \text{Row Total} \\ \hline \text{Men} & 5 & 24 & 21 & 50 \\ \hline \text{Women} & 17 & 21 & 8 & 46 \\ \hline \text{Column Total} & 22 & 45 & 29 & 96 \\ \hline \end{array} \][/tex]
3. Compute the Expected Frequencies:
- The expected frequency for each cell is calculated using the formula:
[tex]\[ E_{ij} = \frac{(\text{Row Total for row } i) \times (\text{Column Total for column } j)}{\text{Grand Total}} \][/tex]
Calculating the expected frequencies for each cell:
- Expected frequency for Men who prefer Sushi ([tex]\(E_{11}\)[/tex]):
[tex]\[ E_{11} = \frac{50 \times 22}{96} \approx 11.5 \][/tex]
- Expected frequency for Men who prefer Pizza ([tex]\(E_{12}\)[/tex]):
[tex]\[ E_{12} = \frac{50 \times 45}{96} \approx 23.4 \][/tex]
- Expected frequency for Men who prefer Hamburgers ([tex]\(E_{13}\)[/tex]):
[tex]\[ E_{13} = \frac{50 \times 29}{96} \approx 15.1 \][/tex]
- Expected frequency for Women who prefer Sushi ([tex]\(E_{21}\)[/tex]):
[tex]\[ E_{21} = \frac{46 \times 22}{96} \approx 10.5 \][/tex]
- Expected frequency for Women who prefer Pizza ([tex]\(E_{22}\)[/tex]):
[tex]\[ E_{22} = \frac{46 \times 45}{96} \approx 21.6 \][/tex]
- Expected frequency for Women who prefer Hamburgers ([tex]\(E_{23}\)[/tex]):
[tex]\[ E_{23} = \frac{46 \times 29}{96} \approx 13.9 \][/tex]
4. Calculate the Chi-Square Test Statistic:
- The formula for the test statistic is:
[tex]\[ \chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}} \][/tex]
Where [tex]\(O_{ij}\)[/tex] is the observed frequency and [tex]\(E_{ij}\)[/tex] is the expected frequency.
Summing these values:
- Contribution from Men who prefer Sushi:
[tex]\[ \frac{(5 - 11.5)^2}{11.5} \approx 3.67 \][/tex]
- Contribution from Men who prefer Pizza:
[tex]\[ \frac{(24 - 23.4)^2}{23.4} \approx 0.02 \][/tex]
- Contribution from Men who prefer Hamburgers:
[tex]\[ \frac{(21 - 15.1)^2}{15.1} \approx 2.31 \][/tex]
- Contribution from Women who prefer Sushi:
[tex]\[ \frac{(17 - 10.5)^2}{10.5} \approx 4.01 \][/tex]
- Contribution from Women who prefer Pizza:
[tex]\[ \frac{(21 - 21.6)^2}{21.6} \approx 0.02 \][/tex]
- Contribution from Women who prefer Hamburgers:
[tex]\[ \frac{(8 - 13.9)^2}{13.9} \approx 2.35 \][/tex]
Adding these contributions gives us:
[tex]\[ \chi^2 \approx 12.4 \][/tex]
5. State the Conclusion:
- The Chi-Square test statistic [tex]\(\chi^2\)[/tex] is approximately 12.4.
Therefore, the computed test statistic, rounded to one decimal place, is [tex]\(\chi^2 = 12.4\)[/tex].
