Questions 6–10 refer to the sample data in the following table, which describes the fate of the passengers and crew aboard the Titanic when it sank on April 15, 1912. Assume that the data are a sample from a large population and we want to use a 0.05 significance level to test the claim that surviving is independent of whether the person is a man, woman, boy, or girl.

What distribution is used to test the stated claim (normal, t, F, chi-square, uniform)?

검증된 단계별 안내

Step 1: Understand the problem. The goal is to test the claim that survival is independent of the category (man, woman, boy, girl). This involves analyzing categorical data, which is presented in a contingency table format.

Step 2: Identify the appropriate statistical test. Since the data involves categorical variables and we are testing for independence, the chi-square test for independence is the appropriate test. This test evaluates whether the observed frequencies in the contingency table differ significantly from the expected frequencies under the assumption of independence.

Step 3: Formulate the null and alternative hypotheses. The null hypothesis (H₀) states that survival is independent of the category (man, woman, boy, girl). The alternative hypothesis (H₁) states that survival is not independent of the category.

Step 4: Calculate the expected frequencies for each cell in the contingency table. Use the formula: E = (row total × column total) / grand total, where E represents the expected frequency for a cell.

Step 5: Compute the chi-square test statistic using the formula: χ² = Σ((O - E)² / E), where O represents the observed frequency and E represents the expected frequency for each cell. Compare the test statistic to the critical value from the chi-square distribution table at the 0.05 significance level to determine whether to reject the null hypothesis.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Chi-Square Test

The Chi-Square test is a statistical method used to determine if there is a significant association between categorical variables. In this context, it helps assess whether survival rates are independent of gender and age categories (men, women, boys, girls) among Titanic passengers. The test compares observed frequencies in each category to expected frequencies, allowing researchers to evaluate the null hypothesis of independence.

Null Hypothesis

The null hypothesis is a statement that assumes no effect or no difference in a statistical test. In this scenario, the null hypothesis posits that survival is independent of the categories of gender and age. Testing this hypothesis allows researchers to determine if any observed differences in survival rates are statistically significant or simply due to random chance.

Significance Level

The significance level, often denoted as alpha (α), is the threshold for determining whether to reject the null hypothesis. A common significance level is 0.05, which indicates a 5% risk of concluding that a difference exists when there is none. In this case, using a 0.05 significance level means that if the p-value from the Chi-Square test is less than 0.05, the null hypothesis of independence would be rejected, suggesting a significant relationship between survival and the categories.