IDNLearn.com connects you with a community of knowledgeable individuals ready to help. Ask any question and receive accurate, in-depth responses from our dedicated team of experts.
Sagot :
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].
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Find precise solutions at IDNLearn.com. Thank you for trusting us with your queries, and we hope to see you again.
