8.4 The Chi-Square Test for Independence

• The null hypothesis (H₀) for the Chi-Square Test for Independence states that the two variables are independent
• The significance level ($\alpha$) in the Chi-Square Test for Independence is typically 0.05
• The randomness condition requires that the data is obtained from a random sample or randomized experiment.

  True
• Expected cell counts in a contingency table must be greater than 5 to ensure the Chi-Square statistic approximates its distribution accurately.

  True
• What is a randomized experiment used for in the Chi-Square Test for Independence?

  Ensuring population representation
• Match the condition with its description:

  Randomness ↔️ Data from random sample

  Independence ↔️ Observations are independent

  Large Counts ↔️ Expected cell counts > 5
• What does the alternative hypothesis state in the Chi-Square Test for Independence?

  Variables are dependent
• What does a p-value less than the significance level indicate in the Chi-Square Test?

  Reject the null hypothesis
• In the Chi-Square Test, the grand total refers to the total number of observations
• All expected cell counts must be greater than 5 for the Chi-Square Test to be valid.


  True
• What does a contingency table display in the Chi-Square Test?

  Joint frequencies of variables
• A contingency table displays the joint frequencies of two categorical variables.
• Contingency tables organize data in a tabular format with categories along rows and columns.


  True
• The row total in the expected frequency formula refers to the sum of observed frequencies in the row containing the cell.
• What is the formula to calculate the Chi-Square statistic ($\chi^{2}$)?


  \chi^{2} = \sum \frac{(Observed - Expected)^{2}}{Expected}</latex>
• If a contingency table has 3 rows and 4 columns, the degrees of freedom are 6.

  True
• What does a p-value of 0.014 suggest if the significance level is 0.05?

  Significant evidence against null
• The test statistic for the Chi-Square Test for Independence is calculated as $\chi^{2} =$$\Sigma ((Observed - Expected)^{2} / Expected)$


  True
• Steps to conduct the Chi-Square Test for Independence

  1️⃣ State the null and alternative hypotheses

  2️⃣ Set the significance level ($\alpha$)

  3️⃣ Create a contingency table

  4️⃣ Calculate expected frequencies

  5️⃣ Calculate the Chi-Square statistic ($\chi^{2}$)

  6️⃣ Determine degrees of freedom

  7️⃣ Find the p-value and make a conclusion
• The independence condition states that observations must be independent of each other
• What does the alternative hypothesis (Hₐ) state in the Chi-Square Test for Independence?

  The two variables are dependent
• Observations in the Chi-Square Test for Independence should be dependent on each other.

  False
• What does the null hypothesis state in the Chi-Square Test for Independence?

  Variables are independent
• The significance level in the Chi-Square Test represents the maximum acceptable risk of a Type I error.


  True
• Steps to create a contingency table:

  1️⃣ Identify categorical variables

  2️⃣ Arrange categories along rows and columns

  3️⃣ Fill in cells with observed frequencies
• What is the purpose of the Chi-Square Test for Independence?

  Assess independence of variables
• What does the null hypothesis state in the Chi-Square Test for Independence?

  Variables are independent
• The most common significance level used in hypothesis testing is 0.05
• What is the purpose of a contingency table?

  Analyze variable relationships
• Steps to create a contingency table:

  1️⃣ Identify categorical variables

  2️⃣ Arrange categories along rows and columns

  3️⃣ Fill cells with observed frequencies
• What is the next step after finding row and column totals when calculating expected frequencies?

  Apply the formula
• The Chi-Square statistic is used to determine if there is evidence to reject the null hypothesis of independence between categorical variables.
• Steps to find the p-value in a Chi-Square Test for Independence:

  1️⃣ Determine degrees of freedom

  2️⃣ Look up the Chi-Square statistic in a table or calculator

  3️⃣ Read the corresponding p-value
• If the p-value is greater than the significance level, we fail to reject the null hypothesis.


  True
• What does the Chi-Square Test for Independence assess?

  Independence of two categorical variables
• How are degrees of freedom calculated in the Chi-Square Test for Independence?

  (Rows - 1) * (Columns - 1)
• What are the three key conditions for using the Chi-Square Test for Independence?

  Randomness, Independence, Large Counts
• What is the minimum value for expected cell counts in the Chi-Square Test for Independence?

  Greater than 5
• The degrees of freedom in the Chi-Square Test for Independence are calculated as (Number of rows - 1) * (Number of columns
• All expected cell counts in the contingency table must be greater than 5