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Questions and Answers
In a Chi-Square test of independence, what does the null hypothesis ($H_0$) typically state?
In a Chi-Square test of independence, what does the null hypothesis ($H_0$) typically state?
- The variables are dependent; changes in one variable cause changes in the other.
- The variables have a strong positive correlation.
- The variables are independent; there is no association between them. (correct)
- The variables have a strong negative correlation.
What does the alternative hypothesis ($H_A$) suggest in a Chi-square test of independence?
What does the alternative hypothesis ($H_A$) suggest in a Chi-square test of independence?
- The observed values are exactly equal to the expected values.
- The variables are independent.
- The variables are dependent. (correct)
- There is no relationship between the variables.
When calculating the Chi-Square test statistic, what does 'O' represent in the formula $\chi^2 = \sum \frac{(O - E)^2}{E}$?
When calculating the Chi-Square test statistic, what does 'O' represent in the formula $\chi^2 = \sum \frac{(O - E)^2}{E}$?
- The observed frequency in a cell of the contingency table (correct)
- The total number of observations
- The critical value from the Chi-Square distribution
- The expected frequency under the null hypothesis
In the Chi-Square test statistic formula $\chi^2 = \sum \frac{(O - E)^2}{E}$, what does 'E' represent?
In the Chi-Square test statistic formula $\chi^2 = \sum \frac{(O - E)^2}{E}$, what does 'E' represent?
What happens to the Chi-Square test statistic if the difference between observed and expected values increases?
What happens to the Chi-Square test statistic if the difference between observed and expected values increases?
How are degrees of freedom (df) calculated in a Chi-Square test of independence for a two-way table?
How are degrees of freedom (df) calculated in a Chi-Square test of independence for a two-way table?
How does the p-value relate to the decision to reject or fail to reject the null hypothesis in a Chi-Square test?
How does the p-value relate to the decision to reject or fail to reject the null hypothesis in a Chi-Square test?
What does a large p-value (e.g., > 0.05) indicate in the context of a Chi-Square test of independence?
What does a large p-value (e.g., > 0.05) indicate in the context of a Chi-Square test of independence?
When interpreting a Chi-Square test, if you fail to reject the null hypothesis, what conclusion can you draw about the relationship between the variables?
When interpreting a Chi-Square test, if you fail to reject the null hypothesis, what conclusion can you draw about the relationship between the variables?
In the 'Popular kids' dataset, what is the primary question being investigated using the Chi-Square test of independence?
In the 'Popular kids' dataset, what is the primary question being investigated using the Chi-Square test of independence?
Using the 'Popular kids' data, the expected count for a cell in the two-way table is calculated using which formula?
Using the 'Popular kids' data, the expected count for a cell in the two-way table is calculated using which formula?
Given the data from the 'Popular kids' dataset, if the calculated Chi-Square statistic is 1.3121 with df = 4, which of the following is the correct p-value interpretation?
Given the data from the 'Popular kids' dataset, if the calculated Chi-Square statistic is 1.3121 with df = 4, which of the following is the correct p-value interpretation?
Based on the 'Popular kids' example, what conclusion is drawn if the p-value is large?
Based on the 'Popular kids' example, what conclusion is drawn if the p-value is large?
What does the Chi-Square test of independence evaluate regarding the relationship between two categorical variables?
What does the Chi-Square test of independence evaluate regarding the relationship between two categorical variables?
If the observed counts in a contingency table are very close to the expected counts, what would you expect the Chi-Square statistic to be?
If the observed counts in a contingency table are very close to the expected counts, what would you expect the Chi-Square statistic to be?
What is the effect of increasing the sample size on the outcome of a Chi-Square test, assuming the effect size remains constant?
What is the effect of increasing the sample size on the outcome of a Chi-Square test, assuming the effect size remains constant?
In a Chi-Square test, what does it mean if the p-value is equal to 0.001?
In a Chi-Square test, what does it mean if the p-value is equal to 0.001?
If you conduct a Chi-Square test and the calculated statistic exceeds the critical value at a predetermined significance level, what is the appropriate conclusion?
If you conduct a Chi-Square test and the calculated statistic exceeds the critical value at a predetermined significance level, what is the appropriate conclusion?
Why is it important to ensure that expected cell counts are not too small when conducting a Chi-Square test?
Why is it important to ensure that expected cell counts are not too small when conducting a Chi-Square test?
The Chi-Square test of independence assumes that the observations are:
The Chi-Square test of independence assumes that the observations are:
In the context of hypothesis testing, what does the significance level (alpha) represent?
In the context of hypothesis testing, what does the significance level (alpha) represent?
What does a contingency table display in the context of a Chi-Square test of independence?
What does a contingency table display in the context of a Chi-Square test of independence?
What type of data is suitable for a Chi-Square test of independence?
What type of data is suitable for a Chi-Square test of independence?
Why is the Chi-Square test considered a non-parametric test?
Why is the Chi-Square test considered a non-parametric test?
If you suspect a causal relationship between two categorical variables, is a Chi-Square test sufficient to establish causality?
If you suspect a causal relationship between two categorical variables, is a Chi-Square test sufficient to establish causality?
Flashcards
Chi-Square Test of Independence
Chi-Square Test of Independence
A statistical test to determine if there is an association between two categorical variables.
Null Hypothesis (H₀) in Chi-Square
Null Hypothesis (H₀) in Chi-Square
The statement that there is no relationship between the two categorical variables being studied.
Alternative Hypothesis (Hₐ) in Chi-Square
Alternative Hypothesis (Hₐ) in Chi-Square
The statement that there is a relationship between the two categorical variables being studied.
Chi-Square Test Statistic
Chi-Square Test Statistic
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Degrees of Freedom (df) for Independence
Degrees of Freedom (df) for Independence
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P-Value in Chi-Square
P-Value in Chi-Square
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Expected Count Formula
Expected Count Formula
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Conclusion when p-value is large
Conclusion when p-value is large
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Study Notes
Chi-Square Test of Independence
- It is used to determine if there is a significant association between two categorical variables
Popular Kids Data Set
- Students in grades 4-6 were surveyed on whether good grades, athletic ability, or popularity was most important to them
- A two-way table separates students by grade and by their choice of the most important factor
Hypotheses
- Null hypothesis (H0): Grade and goals are independent, Goals do not vary by grade
- Alternative hypothesis (HA): Grade and goals are dependent, Goals vary by grade
Test Statistic
- χ2df = Σ [(O – E)² / E]
- df = (R − 1) × (C − 1)
- Where:
- k is the number of cells
- R is the number of rows
- C is the number of columns
- df is calculated differently for one-way and two-way tables
P-Value
- It is the area under the χ2df curve, above the calculated test statistic
Expected Counts in Two-Way Tables
- Expected Count = (row total) × (column total) / table total
- For example, given the totals:
- Grades: 247
- Popular: 141
- Sports: 90
- 4th: 119
- 5th: 176
- 6th: 183
- Total: 478
- Erow 1,col 1 = (119 x 247) / 478 = 61
- Erow 1,col 2 = (119 × 141) / 478 = 35
- 176 x 141 / 478 = 52, more than the expected number of 5th graders have a goal of being popular
Calculating the Test Statistic
- Expected counts have been calculated
- x² = Σ (63-61)² / 61 + (31-35)² / 35 +……+ (32 – 34)² / 34 = 1.3121
- df = (R – 1) × (C – 1) = (3 – 1) × (3 – 1) = 2 × 2 = 4
Calculating the P-Value
- For X2df = 1.3121 and df = 4, the p-value is more than 0.3
Conclusion
- The p-value is large
- Fail to reject H0
- The data does not provide convincing evidence that grade and goals are dependent
- It doesn't appear that goals vary by grade
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