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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 influence each other.
- The variables are dependent.
- The variables are independent. (correct)
- There is a significant association between the variables.
Which of the following is an assumption for the Chi-Square test of independence to be valid?
Which of the following is an assumption for the Chi-Square test of independence to be valid?
- The observations are paired.
- The expected cell counts are less than 1.
- The data follows a normal distribution.
- The observations are independent. (correct)
What does a small p-value (e.g., p < 0.05) in a Chi-Square Test of Independence suggest?
What does a small p-value (e.g., p < 0.05) in a Chi-Square Test of Independence suggest?
- Failure to reject the null hypothesis.
- The sample size is too small.
- There is strong evidence against the null hypothesis. (correct)
- The variables are independent.
Which formula is used to calculate the degrees of freedom (df) in a Chi-Square Test of Independence for a two-way table?
Which formula is used to calculate the degrees of freedom (df) in a Chi-Square Test of Independence for a two-way table?
In the context of the 'popular kids' dataset, what does rejecting the null hypothesis imply?
In the context of the 'popular kids' dataset, what does rejecting the null hypothesis imply?
The Chi-Square test statistic is calculated by summing the squared differences between observed and expected values, divided by the expected values. What does a larger test statistic indicate?
The Chi-Square test statistic is calculated by summing the squared differences between observed and expected values, divided by the expected values. What does a larger test statistic indicate?
If the calculated Chi-Square test statistic is 1.3121 with df = 4, how does this influence the conclusion of the hypothesis test?
If the calculated Chi-Square test statistic is 1.3121 with df = 4, how does this influence the conclusion of the hypothesis test?
Why is it important to calculate expected counts in a Chi-Square Test of Independence?
Why is it important to calculate expected counts in a Chi-Square Test of Independence?
What does the alternative hypothesis ($H_A$) state in the context of the 'popular kids' dataset?
What does the alternative hypothesis ($H_A$) state in the context of the 'popular kids' dataset?
How does the Chi-Square Test of Independence assist in drawing conclusions from the popular kids survey data?
How does the Chi-Square Test of Independence assist in drawing conclusions from the popular kids survey data?
Using the 'popular kids' dataset, if the p-value is greater than 0.05, what conclusion can be drawn?
Using the 'popular kids' dataset, if the p-value is greater than 0.05, what conclusion can be drawn?
What is the purpose of performing a Chi-Square Test of Independence?
What is the purpose of performing a Chi-Square Test of Independence?
In the formula for the Chi-Square test statistic, what do 'O' and 'E' represent?
In the formula for the Chi-Square test statistic, what do 'O' and 'E' represent?
If a Chi-Square Test of Independence showed a statistically significant association between student grades and preference for athletic ability, how could this be interpreted?
If a Chi-Square Test of Independence showed a statistically significant association between student grades and preference for athletic ability, how could this be interpreted?
Why is the p-value compared against a significance level (alpha) in a Chi-Square Test of Independence?
Why is the p-value compared against a significance level (alpha) in a Chi-Square Test of Independence?
What is the expected count?
What is the expected count?
Imagine a scenario where you're analyzing survey data about favorite subjects among high school students from different grades (9-12). If the Chi-Square test reveals that a student's grade level and their favorite subject are independent, how should this be interpreted?
Imagine a scenario where you're analyzing survey data about favorite subjects among high school students from different grades (9-12). If the Chi-Square test reveals that a student's grade level and their favorite subject are independent, how should this be interpreted?
In a study examining whether there is a relationship between smoking habits (smoker, non-smoker) and the incidence of lung disease (yes, no), a Chi-Square Test of Independence is conducted. Which of the following statements correctly interprets the null hypothesis in this context?
In a study examining whether there is a relationship between smoking habits (smoker, non-smoker) and the incidence of lung disease (yes, no), a Chi-Square Test of Independence is conducted. Which of the following statements correctly interprets the null hypothesis in this context?
Suppose a researcher is using a Chi-Square test to analyze whether political affiliation (Democrat, Republican, Independent) is related to support for a particular policy (support, oppose, abstain). The calculated Chi-Square statistic is very large, and the resulting p-value is close to zero. What is the most appropriate conclusion?
Suppose a researcher is using a Chi-Square test to analyze whether political affiliation (Democrat, Republican, Independent) is related to support for a particular policy (support, oppose, abstain). The calculated Chi-Square statistic is very large, and the resulting p-value is close to zero. What is the most appropriate conclusion?
A marketing team wants to know if there's a link between the region a customer lives in (North, South, East, West) and their preferred method of communication (email, phone, mail). They conduct a Chi-Square test and find a p-value of 0.15. What action should the marketing team take based on this result?
A marketing team wants to know if there's a link between the region a customer lives in (North, South, East, West) and their preferred method of communication (email, phone, mail). They conduct a Chi-Square test and find a p-value of 0.15. What action should the marketing team take based on this result?
What is the main reason for using the Chi-Square test of independence instead of other statistical tests like t-tests or ANOVA?
What is the main reason for using the Chi-Square test of independence instead of other statistical tests like t-tests or ANOVA?
In the context of hypothesis testing, failing to reject the null hypothesis implies:
In the context of hypothesis testing, failing to reject the null hypothesis implies:
When calculating the expected counts for a cell in a contingency table, what does the formula (row total) x (column total) / (table total) represent?
When calculating the expected counts for a cell in a contingency table, what does the formula (row total) x (column total) / (table total) represent?
Imagine a food scientist is testing whether consumers' preferences for different types of snacks (sweet, salty, savory) are related to their age group (young, middle-aged, senior). Upon conducting a Chi-Square test of independence, what type of data should the scientist have collected?
Imagine a food scientist is testing whether consumers' preferences for different types of snacks (sweet, salty, savory) are related to their age group (young, middle-aged, senior). Upon conducting a Chi-Square test of independence, what type of data should the scientist have collected?
Flashcards
Chi-Square Test of Independence
Chi-Square Test of Independence
A statistical test used to determine if there is a significant association between two categorical variables.
Null Hypothesis (H₀)
Null Hypothesis (H₀)
The statement assuming no association between the variables being tested. Any difference is due to chance.
Alternative Hypothesis (Hₐ)
Alternative Hypothesis (Hₐ)
The statement suggesting an association between the variables. The goals vary by grade.
Chi-Square Test Statistic Formula
Chi-Square Test Statistic Formula
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Degrees of Freedom (df) for Independence Test
Degrees of Freedom (df) for Independence Test
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P-value
P-value
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Expected Count (in two-way tables)
Expected Count (in two-way tables)
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Conclusion: High P-value
Conclusion: High P-value
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Study Notes
Chi-Square Test of Independence
- This test can be applied to data about goals of students from grades 4-6, specifically good grades, athletic ability, or popularity
- The data is displayed in a two-way table separating the students by grade and by choice of most important factor
Hypotheses
- H0 (Null Hypothesis): Grade and goals are independent, meaning goals do not vary by grade
- HA (Alternative Hypothesis): Grade and goals are dependent, meaning goals vary by grade
Test Statistic
- To calculate the test statistic: χ2df = ∑ ((O – E)2 / E)
- df = (R − 1) × (C − 1)
- k = the number of cells
- R = the number of rows
- C = the number of columns
- The degrees of freedom are calculated differently for one-way and two-way tables
- The p-value is the area under the χ2df curve, above the calculated test statistic
Expected Counts in Two-Way Tables
- The formula to calculate the expected count: Expected Count = (row total) × (column total) / table total
- For example, given row 1 total = 119, column 1 total = 247, and table total = 478: Erow 1,col 1 = (119 x 247) / 478 = 61
- Observed count for a highlighted cell = 55
- The expected count for this highlighted cell is (176 x 141) / 478 = 52, which shows more than expected number of 5th graders have a goal of being popular
Calculating the Test Statistic in Two-Way Tables
- Expected counts are often displayed in blue next to the observed counts to ease calculation
- Example calculation: χ2 =∑ ((63 − 61)2 / 61) + ((31 − 35)2 / 35) +…+ ((32 – 34)2 / 34) = 1.3121
- Where df = (R – 1) × (C – 1) = (3 – 1) × (3 – 1) = 2 × 2 = 4
Calculating the P-Value
- Given χ2df = 1.3121 and df = 4, the p-value for the hypothesis test is more than 0.3
Conclusion
- Since the p-value is large, the null hypothesis (H0) is not rejected
- The data does not provide convincing evidence that grade and goals are dependent, meaning goals do not appear to vary by grade
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