To complete this project, I will follow the steps outlined and provide a clear structure. Here's how I will proceed:

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Step 1: Pick a Bivariate Data Set

I will choose (Favorite Music Genre; Preferred Beverage) for a sample size of 50 participants. This ensures a diverse dataset and makes the analysis more interesting.

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Step 2: Observed Count Table

I will collect data where the two categorical variables intersect. For example:

Music Genre/ Beverage	Coffee	Tea	Juice	Soda	Total
Pop	6	8	5	7	26
Rock	4	5	3	2	14
Jazz	3	1	2	4	10
Total	13	14	10	13	50

This is our Observed Count Table.

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Step 3: Calculate the Expected Count Table

The expected counts ($E$) are calculated using: $E_{ij} = \frac{(Row \ Total \times Column \ Total)}{\text{Grand Total}}$

For each cell:

• $E_{Pop, Coffee} = \frac{(26 \times 13)}{50} = 6.76$
• Repeat for each cell.

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Step 4: Calculate the Deviations Table

The deviations ($(O - E)^2/E$) will be calculated for each cell using: $\chi^2 = \sum \frac{(O - E)^2}{E}$

For example:

• $Deviation_{Pop, Coffee} = \frac{(6 - 6.76)^2}{6.76}$

Sum these deviations for all cells.

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Step 5: Calculate Degrees of Freedom

The degrees of freedom ($df$) for a contingency table are: $df = (Rows - 1) \times (Columns - 1)$ For this table: $df = (3 - 1) \times (4 - 1) = 6$

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Step 6: Calculate P-Value

Using an online tool (like Bognar’s app), input $\chi^2$ and $df$ to calculate the P-value. The results will determine if the null hypothesis is rejected.

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Step 7: Discuss and Test the Independence Hypothesis

• Null Hypothesis ($H_0$): The two variables are independent.
• Alternative Hypothesis ($H_a$): The two variables are not independent.
• Significance Level ($\alpha$): 0.05

If the P-value $\leq$ 0.05, reject $H_0$. Otherwise, fail to reject $H_0$.

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Would you like me to proceed with the calculations or clarify any part of this process? Let me know!

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Related Questions:

1. What other examples of bivariate categorical datasets could be analyzed?
2. How do changes in sample size affect the chi-square test?
3. What is the impact of larger contingency tables on degrees of freedom?
4. Can this test be applied to more than two variables at a time? How?
5. What are the limitations of the chi-square test for independence?

Tip:

Always ensure observed counts are sufficiently large ($\geq 5$) in each cell to validate the chi-square test results.