Identifying Hypotheses Refer to the data given in Exercise 1 and assume that the requirements are all satisfied and we want to conduct a hypothesis test of independence using the methods of this section. Identify the null and alternative hypotheses.

Verified step by step guidance

1. Step 1: Understand the context of the problem. A hypothesis test of independence is used to determine whether two categorical variables are independent or associated. Independence means that the occurrence of one variable does not affect the occurrence of the other.
2. Step 2: Define the null hypothesis (H₀). The null hypothesis in a test of independence states that the two variables are independent. In mathematical terms, this can be expressed as: $H₀:\; P(A\; \cap \; B)\; =\; P(A)\; \times \; P(B)$, where A and B are the two categorical variables.
3. Step 3: Define the alternative hypothesis (H₁). The alternative hypothesis states that the two variables are not independent, meaning there is an association between them. In mathematical terms, this can be expressed as: $H₁:\; P(A\; \cap \; B)\; \ne \; P(A)\; \times \; P(B)$.
4. Step 4: Ensure that the requirements for conducting the test are satisfied. These typically include having a sufficiently large sample size and ensuring that the expected frequencies in each cell of the contingency table are at least 5.
5. Step 5: Prepare to use the chi-square test statistic to evaluate the hypotheses. The test statistic is calculated using the formula: $\chi ²\; =\; \Sigma ((O\; -\; E)²\; /\; E)$, where O represents the observed frequencies and E represents the expected frequencies under the assumption of independence.

Key Concepts

Null Hypothesis

The null hypothesis (H0) is a statement that indicates no effect or no difference in the context of a statistical test. It serves as a default position that assumes any observed differences in data are due to random chance. In hypothesis testing, the goal is to gather evidence to either reject or fail to reject the null hypothesis based on sample data.

Alternative Hypothesis

The alternative hypothesis (H1 or Ha) is a statement that contradicts the null hypothesis, suggesting that there is an effect or a difference. It represents the researcher's claim or the outcome they are trying to prove. In hypothesis testing, if the evidence is strong enough to reject the null hypothesis, the alternative hypothesis is considered supported.

Hypothesis Test of Independence

A hypothesis test of independence is used to determine whether there is a significant association between two categorical variables. This test evaluates whether the distribution of one variable differs across the levels of another variable. Commonly, the Chi-square test is employed for this purpose, allowing researchers to assess the relationship between the variables based on observed and expected frequencies.