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

Triola - Elementary Statistics 14th Edition

Triola14th EditionElementary StatisticsISBN: 9780137366446Not the one you use?Change textbook

Triola 14th Edition

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

Problem 11.q.1

Chapter 11, Problem 11.q.1

Exercises 1–5 refer to the sample data in the following table, which summarizes the frequencies of 500 digits randomly generated by Statdisk. Assume that we want to use a 0.05 significance level to test the claim that Statdisk generates the digits in a way that they are equally likely.

What are the null and alternative hypotheses corresponding to the stated claim?

Verified step by step guidance

Step 1: Understand the problem. The goal is to test the claim that Statdisk generates digits in a way that they are equally likely. This involves setting up hypotheses for a chi-square goodness-of-fit test.

Step 2: Define the null hypothesis (H₀). The null hypothesis states that the digits are equally likely to occur, meaning the expected frequency for each digit is the same. Mathematically, H₀: P(0) = P(1) = P(2) = ... = P(9).

Step 3: Define the alternative hypothesis (H₁). The alternative hypothesis states that the digits are not equally likely to occur, meaning at least one digit has a different probability. Mathematically, H₁: At least one P(i) ≠ P(j) for i ≠ j.

Step 4: Calculate the expected frequency for each digit under the null hypothesis. Since there are 500 total digits and 10 possible digits, the expected frequency for each digit is 500 ÷ 10 = 50.

Step 5: Use the observed frequencies from the table and the expected frequencies to compute the chi-square test statistic. The formula for the chi-square test statistic is χ² = Σ((Oᵢ - Eᵢ)² / Eᵢ), where Oᵢ is the observed frequency and Eᵢ is the expected frequency for each digit.

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Key Concepts

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

Null Hypothesis (H0)

The null hypothesis is a statement that assumes no effect or no difference, serving as a default position in hypothesis testing. In this context, it posits that the digits generated by Statdisk are equally likely, meaning each digit from 0 to 9 has the same probability of occurrence. This hypothesis is tested against the alternative hypothesis to determine if there is enough evidence to reject it.

Alternative Hypothesis (H1)

The alternative hypothesis is a statement that contradicts the null hypothesis, suggesting that there is an effect or a difference. For this scenario, the alternative hypothesis would claim that the digits generated by Statdisk are not equally likely, indicating that some digits occur more frequently than others. This hypothesis is what researchers aim to support through statistical testing.

Significance Level (α)

The significance level, denoted as alpha (α), is the threshold for determining whether to reject the null hypothesis. In this case, a significance level of 0.05 indicates that there is a 5% risk of concluding that a difference exists when there is none. If the p-value obtained from the statistical test is less than 0.05, the null hypothesis would be rejected in favor of the alternative hypothesis.

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Textbook Question

Exercises 1–5 refer to the sample data in the following table, which summarizes the frequencies of 500 digits randomly generated by Statdisk. Assume that we want to use a 0.05 significance level to test the claim that Statdisk generates the digits in a way that they are equally likely.

Is the hypothesis test left-tailed, right-tailed, or two-tailed?

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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.

b. Identify the observed and expected values for the leading digit of 2.

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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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c. Using the probabilities found in part (b), find the expected frequency for each category.

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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.

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

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Textbook Question

Is the hypothesis test left-tailed, right-tailed, or two-tailed?

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Textbook Question

c. Use the results from part (b) to find the contribution to the x2 test statistic from the category representing the leading digit of 2.

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