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Ch. 10 - Chi-Square Tests and the F-Distribution
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 10 - Chi-Square Tests and the F-DistributionProblem 10.1.10d
Chapter 10, Problem 10.1.10d

Performing a Chi-Square Goodness-of-Fit Test
In Exercises 7–16, (d) decide whether to reject or fail to reject the null hypothesis.


Ways to Pay A financial analyst claims that the distribution of people’s preferences on how to pay for goods is different from the distribution shown in the figure. You randomly select 600 people and record their preferences on how to pay for goods. The table shows the results. At α=0.01, test the financial analyst’s claim. (Adapted from Travis Credit Union)


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Step 1: State the null and alternative hypotheses. The null hypothesis (H₀) is that the distribution of people's preferences for payment methods matches the distribution shown in the figure. The alternative hypothesis (H₁) is that the distribution of people's preferences for payment methods is different from the distribution shown in the figure.
Step 2: Calculate the expected frequencies for each payment method based on the percentages provided in the figure and the total sample size of 600 people. Use the formula: Expected frequency = (Percentage from figure / 100) × Total sample size. For example, for 'Cash', the expected frequency is (29 / 100) × 600.
Step 3: Compute the Chi-Square test statistic using the formula: χ² = Σ((Observed frequency - Expected frequency)² / Expected frequency). For each payment method, subtract the expected frequency from the observed frequency, square the result, divide by the expected frequency, and sum these values across all categories.
Step 4: Determine the degrees of freedom (df) for the test. The degrees of freedom are calculated as: df = Number of categories - 1. In this case, there are 4 payment categories, so df = 4 - 1 = 3.
Step 5: Compare the calculated Chi-Square test statistic to the critical value from the Chi-Square distribution table at α = 0.01 and df = 3. If the test statistic exceeds the critical value, reject the null hypothesis; otherwise, fail to reject the null hypothesis.

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Key Concepts

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

Chi-Square Goodness-of-Fit Test

The Chi-Square Goodness-of-Fit Test is a statistical method used to determine if the observed frequencies of a categorical variable differ significantly from the expected frequencies. It compares the actual data collected from a sample to a theoretical distribution, allowing researchers to assess whether a specific hypothesis about the population is supported by the sample data.
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Goodness of Fit Test

Null Hypothesis

The null hypothesis (H0) is a statement that assumes no effect or no difference in a statistical test. In the context of the Chi-Square Goodness-of-Fit Test, it posits that the observed distribution of preferences for payment methods matches the expected distribution. Rejecting the null hypothesis suggests that there is a significant difference between the observed and expected frequencies.
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Step 1: Write Hypotheses

Significance Level (α)

The significance level (α) is a threshold used in hypothesis testing to determine whether to reject the null hypothesis. It represents the probability of making a Type I error, which occurs when the null hypothesis is incorrectly rejected. In this case, α is set at 0.01, indicating a 1% risk of concluding that a difference exists when there is none, thus requiring strong evidence to reject the null hypothesis.
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Related Practice
Textbook Question

Performing a Chi-Square Goodness-of-Fit Test

In Exercises 7–16, (b) find the critical value and identify the rejection region.


Ways to Pay A financial analyst claims that the distribution of people’s preferences on how to pay for goods is different from the distribution shown in the figure. You randomly select 600 people and record their preferences on how to pay for goods. The table shows the results. At α=0.01, test the financial analyst’s claim. (Adapted from Travis Credit Union)


49
views
Textbook Question

Performing a Chi-Square Goodness-of-Fit Test

In Exercises 7–16, (a) identify the claim and state H₀ and Hₐ.


Ways to Pay A financial analyst claims that the distribution of people’s preferences on how to pay for goods is different from the distribution shown in the figure. You randomly select 600 people and record their preferences on how to pay for goods. The table shows the results. At α=0.01, test the financial analyst’s claim. (Adapted from Travis Credit Union)


71
views
Textbook Question

Performing a Chi-Square Goodness-of-Fit Test

In Exercises 7–16, (c) find the chi-square test statistic.


Ways to Pay A financial analyst claims that the distribution of people’s preferences on how to pay for goods is different from the distribution shown in the figure. You randomly select 600 people and record their preferences on how to pay for goods. The table shows the results. At α=0.01, test the financial analyst’s claim. (Adapted from Travis Credit Union)


67
views
Textbook Question

List the two conditions that must be met in order to use the paired-sample sign test.

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