### Question 13 ### Understanding Chi-Square Goodness-of-Fit Test and p-ValuesA \( \chi^2 \) (chi-square) goodness-of-fit test is used to determine whether a sample data matches a population with a specific distribution. Here, we have an example where all assumptions for the test are met, resulting in a chi-square test statistic \( \chi^2 = 1.92 \) and a corresponding p-value of 0.75.The researcher's interpretation that the p-value represents a 0.75 probability of observing a test statistic of \( \chi^2 = 1.92 \) or larger is under scrutiny. The concern is identifying what might be wrong with this interpretation.#### What's Wrong with the Researcher's Interpretation?- **Option A:** The researcher did not state that the p-value is conditional on the null hypothesis being true.- **Option B:** The researcher interpreted the p-value as the probability of observing 1.92 exactly.- **Option C:** The alternative hypothesis is not stated.- **Option D:** The significance level is not stated.- **Option E:** The degrees of freedom are not stated.To correct the interpretation, it is critical to understand the definition and implications of the p-value:1. **p-Value Definition:** The p-value measures the probability of obtaining test results at least as extreme as the results actually observed, assuming that the null hypothesis is true. 2. **Common Misinterpretations:** - **Exact Value Fallacy:** A p-value is not the probability of observing an exact test statistic (e.g., \( \chi^2 = 1.92 \)), but rather the probability of observing a test statistic as extreme or more extreme, given the null hypothesis. - **Conditional Probability:** The p-value is conditional on the null hypothesis being true, which many fail to explicitly state.#### Detailed Explanation of Options- **Option A (Correct)**: The researcher must note that the p-value's validity hinges on the assumption that the null hypothesis is true, which was neglected.- **Option B (Correct)**: The specific error here is interpreting the p-value as the probability of getting exactly 1.92, rather than 1.92 or something more extreme.- **Option C (Not Correct)**: While omitting the alternative hypothesis can be problematic, it doesn't directly impact the p-value interpretation.- **Option D (Not Correct)**: Though reporting the

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A major credit card company is investigating whether the distribution of the number of credit cards used by its customers has changed from last year to this year. Customers are classified as using 1 card, 2 cards, or more than 2 cards. The company conducts a chi-square goodness-of-fit test to investigate whether there is a change in the distribution of number of cards used from last year to this year. The value of the chi-square test statistic was x? = 7.82 with a corresponding p-value of 0.02. Assuming the conditions for inference were met, which of the following is the correct interpretation of this p-value? There is a 2 percent chance that the company's claim is correct. There is a 2 percent chance of obtaining a chi-square value of at least 7.82. If the null hypothesis were true, there is a 2 percent chance of obtaining a chi-square value of at least 7.82. If the null hypothesis were true, there is a 2 percent chance that the company's claim is correct. E If the null hypothesis were true, there is a 2 percent chance of obtaining a chi-square value of 7.82.

A major credit card company is investigating whether the distribution of the number of credit cards used by its customers has changed from last year to this year. Customers are classified as using 1 card, 2 cards, or more than 2 cards. The company conducts a chi-square goodness-of-fit test to investigate whether there is a change in the distribution of number of cards used from last year to this year. The value of the chi-square test statistic was x? = 7.82 with a corresponding p-value of 0.02. Assuming the conditions for inference were met, which of the following is the correct interpretation of this p-value? There is a 2 percent chance that the company's claim is correct. There is a 2 percent chance of obtaining a chi-square value of at least 7.82. If the null hypothesis were true, there is a 2 percent chance of obtaining a chi-square value of at least 7.82. If the null hypothesis were true, there is a 2 percent chance that the company's claim is correct. E If the null hypothesis were true, there is a 2 percent chance of obtaining a chi-square value of 7.82.

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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