Ch. 11 - Goodness-of-Fit and Contingency Tables

Triola - Elementary Statistics 14th Edition

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Triola 14th Edition

Ch. 11 - Goodness-of-Fit and Contingency Tables

문제 11.CQQ.10

11장, 문제 11.CQQ.10

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.

Given that the P-value for the hypothesis test is 0.000 when rounded to three decimal places, what do you conclude? What do the results indicate about the rule that women and children should be the first to be saved?

검증된 단계별 안내

Step 1: Identify the hypothesis test being conducted. The null hypothesis (H₀) is that survival is independent of the category (man, woman, boy, girl). The alternative hypothesis (H₁) is that survival is dependent on the category.

Step 2: Recognize the type of test used. This is a chi-square test for independence, as we are testing the relationship between two categorical variables: survival status and category (man, woman, boy, girl).

Step 3: Analyze the P-value provided. The P-value is 0.000 (rounded to three decimal places), which is less than the significance level of 0.05. This indicates strong evidence against the null hypothesis.

Step 4: Conclude based on the P-value. Since the P-value is less than 0.05, we reject the null hypothesis. This means that survival is not independent of the category (man, woman, boy, girl).

Step 5: Interpret the results in context. The results suggest that survival was influenced by the category, supporting the idea that women and children were prioritized for rescue, as indicated by the higher survival rates for women and girls compared to men and boys.

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주요 개념

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

Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about a population based on sample data. It involves formulating a null hypothesis (H0) that represents no effect or no difference, and an alternative hypothesis (H1) that indicates the presence of an effect. The test evaluates the evidence against H0 using a significance level (alpha), typically set at 0.05, to determine whether to reject or fail to reject the null hypothesis.

P-value

The P-value is a measure that helps determine the strength of the evidence against the null hypothesis in hypothesis testing. It represents the probability of observing the sample data, or something more extreme, if the null hypothesis is true. A low P-value (typically less than 0.05) indicates strong evidence against H0, leading to its rejection, while a high P-value suggests insufficient evidence to reject H0.

Independence in Statistics

Independence in statistics refers to the scenario where the occurrence of one event does not affect the probability of another event. In the context of the Titanic data, testing for independence involves examining whether survival rates are influenced by gender or age. If the null hypothesis of independence is rejected, it suggests that survival is significantly associated with these factors, challenging the notion that women and children were prioritized in rescue efforts.

관련 실천

교과서 질문

Testing Goodness-of-Fit with a Normal Distribution Refer to Data Set 1 “Body Data” in Appendix B for the heights of females.

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a. Enter the observed frequencies in the table above.

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교과서 질문

In Exercises 5–20, conduct the hypothesis test and provide the test statistic and the P-value and/or critical value, and state the conclusion.

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교과서 질문

In Exercises 5–20, conduct the hypothesis test and provide the test statistic and the P-value and/or critical value, and state the conclusion.

Testing a Slot Machine The author purchased a slot machine (Bally Model 809) and tested it by playing it 1197 times. There are 10 different categories of outcomes, including no win, win jackpot, win with three bells, and so on. When testing the claim that the observed outcomes agree with the expected frequencies, the author obtained a test statistic of x2 = 8.815 Use a 0.05 significance level to test the claim that the actual outcomes agree with the expected frequencies. Does the slot machine appear to be functioning as expected?

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교과서 질문

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.

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Does the Treatment Affect Success? The following table lists frequencies of successes and failures for different treatments used for a stress fracture in a foot bone (based on data from “Surgery Unfounded for Tarsal Navicular Stress Fracture,” by Bruce Jancin, Internal Medicine News, Vol. 42, No. 14). Use a 0.05 significance level to test the claim that success of the treatment is independent of the type of treatment. What does the result indicate about the increasing trend to use surgery?

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교과서 질문

Weather-Related Deaths For the most recent year as of this writing, the numbers of weather-related U.S. deaths for each month were 61, 14, 22, 26, 29, 42, 93, 49, 47, 35, 96, 16, listed in order beginning with January (based on data from the National Weather Service). Use a 0.01 significance level to test the claim that weather-related deaths occur in the different months with the same frequency. Provide an explanation for the result.

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