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

Problem 11.RE.1

Chapter 11, Problem 11.RE.1

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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Step 1: Identify the hypothesis. The null hypothesis (H₀) states that weather-related deaths occur with the same frequency across all months. The alternative hypothesis (H₁) states that weather-related deaths do not occur with the same frequency across all months.

Step 2: Choose the appropriate test. Since we are comparing observed frequencies (the number of deaths in each month) to expected frequencies (assuming equal distribution), we use the chi-square goodness-of-fit test.

Step 3: Calculate the expected frequency for each month. Assuming equal distribution, divide the total number of deaths by the number of months. Use the formula: Expected frequency = Total deaths / 12.

Step 4: Compute the chi-square test statistic. Use the formula: χ² = Σ((Oᵢ - Eᵢ)² / Eᵢ), where Oᵢ is the observed frequency for each month, and Eᵢ is the expected frequency. Perform this calculation for all 12 months and sum the results.

Step 5: Compare the test statistic to the critical value. Determine the degrees of freedom (df = number of categories - 1 = 12 - 1 = 11) and find the critical value for a 0.01 significance level from the chi-square distribution table. If the test statistic exceeds the critical value, reject the null hypothesis; otherwise, fail to reject it.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 or difference. In this context, the null hypothesis would state that weather-related deaths occur with the same frequency across all months, while the alternative would suggest that they do not.

Chi-Square Test

The Chi-Square test is a statistical test used to determine if there is a significant association between categorical variables. In this scenario, it can be applied to assess whether the observed frequencies of weather-related deaths in each month differ significantly from the expected frequencies if deaths were uniformly distributed. The test calculates a Chi-Square statistic, which is then compared to a critical value from the Chi-Square distribution to determine significance.

Significance Level

The significance level, often denoted as alpha (α), is the threshold for determining whether to reject the null hypothesis. In this case, a significance level of 0.01 indicates that there is a 1% risk of concluding that a difference exists when there is none. If the p-value obtained from the Chi-Square test is less than 0.01, it suggests that the differences in weather-related deaths across months are statistically significant, warranting further investigation.

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Step 4: State Conclusion Example 4

Related Practice

Textbook Question

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?

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Textbook Question

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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Textbook Question

Cybersecurity The table below lists the frequency of leading digits of Internet traffic interarrival times for a computer, along with the percentages of each leading digit expected with Benford’s law.

a. Identify the general notation used for observed and expected values.

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Textbook Question

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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Textbook Question

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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Textbook Question

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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b. Assuming a normal distribution with mean and standard deviation given by the sample mean and standard deviation, use the methods of Chapter 6 to find the probability of a randomly selected height belonging to each class.

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