Math Problem Statement
Test the independence of the two variables in the table below: Enter the test statistic (to 4 decimal places)
Var2 below\Var1->
Yes
No
Happy
40
80
Sad
120
200
Solution
To test the independence of the two variables in a contingency table, we use a chi-square test of independence. Here's how to calculate the test statistic:
1. Given Table:
& \textbf{Yes} & \textbf{No} & \textbf{Total} \\ \hline \textbf{Happy} & 40 & 80 & 120 \\ \textbf{Sad} & 120 & 200 & 320 \\ \hline \textbf{Total} & 160 & 280 & 440 \\ \end{array}$$ ### 2. Expected Frequencies The expected frequency for each cell is calculated as: $$E_{ij} = \frac{(\text{Row Total}) \cdot (\text{Column Total})}{\text{Grand Total}}$$ - $$E_{\text{Happy, Yes}} = \frac{120 \cdot 160}{440} = 43.6364$$ - $$E_{\text{Happy, No}} = \frac{120 \cdot 280}{440} = 76.3636$$ - $$E_{\text{Sad, Yes}} = \frac{320 \cdot 160}{440} = 116.3636$$ - $$E_{\text{Sad, No}} = \frac{320 \cdot 280}{440} = 203.6364$$ ### 3. Chi-Square Test Statistic Formula $$\chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}$$ Where $$O_{ij}$$ is the observed frequency, and $$E_{ij}$$ is the expected frequency. #### For each cell: - $$\chi^2_{\text{Happy, Yes}} = \frac{(40 - 43.6364)^2}{43.6364} = 0.3033$$ - $$\chi^2_{\text{Happy, No}} = \frac{(80 - 76.3636)^2}{76.3636} = 0.1754$$ - $$\chi^2_{\text{Sad, Yes}} = \frac{(120 - 116.3636)^2}{116.3636} = 0.1165$$ - $$\chi^2_{\text{Sad, No}} = \frac{(200 - 203.6364)^2}{203.6364} = 0.0650$$ ### 4. Sum of Chi-Square Values: $$\chi^2 = 0.3033 + 0.1754 + 0.1165 + 0.0650 = 0.6602$$ ### Final Answer: The test statistic is **0.6602** (rounded to 4 decimal places). Would you like to discuss the next steps, such as finding the p-value? Here are some related questions: --- 1. How do you determine the degrees of freedom for this test? 2. How is the critical value of chi-square determined from a table? 3. What is the significance of the p-value in the chi-square test? 4. When can the chi-square test not be applied due to small expected frequencies? 5. How do you interpret the result if the null hypothesis is rejected? **Tip:** Always ensure your observed and expected frequencies meet the chi-square test assumptions—expected values should generally be ≥ 5.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Statistics
Chi-Square Test of Independence
Formulas
Chi-Square Test Statistic Formula: χ² = Σ [(Oij - Eij)² / Eij]
Expected Frequency Formula: Eij = (Row Total * Column Total) / Grand Total
Theorems
Chi-Square Test of Independence
Suitable Grade Level
Grades 11-12
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