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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 10.1.10c
Chapter 10, Problem 10.1.10c

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)


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Step 1: Understand the problem. The Chi-Square Goodness-of-Fit Test is used to determine whether the observed frequencies differ significantly from the expected frequencies. Here, the expected distribution is given in percentages (Cash: 29%, Debit or Credit: 59%, Check: 5%, Digital Wallet/Other: 7%), and the observed frequencies are provided in the table (Cash: 194, Debit or Credit: 338, Check: 21, Digital Wallet/Other: 47). The total sample size is 600 people.
Step 2: Calculate the expected frequencies for each category. Multiply the total sample size (600) by the percentage for each category to find the expected frequency. For example, for Cash, the expected frequency is calculated as: 600×0.29. Repeat this for Debit or Credit, Check, and Digital Wallet/Other.
Step 3: Compute the Chi-Square test statistic for each category using the formula: Ω=(Oi-Ei)²Ei, where Oi is the observed frequency and Ei is the expected frequency for each category. Perform this calculation for Cash, Debit or Credit, Check, and Digital Wallet/Other.
Step 4: Sum the Chi-Square test statistics for all categories to obtain the overall Chi-Square test statistic. This value represents the total deviation between observed and expected frequencies.
Step 5: Compare the calculated Chi-Square test statistic to the critical value from the Chi-Square distribution table at α=0.01 and degrees of freedom (df = number of categories - 1). If the test statistic exceeds the critical value, reject the null hypothesis and conclude that the distribution of preferences is significantly different from the expected distribution.

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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 and Alternative Hypothesis

In hypothesis testing, the null hypothesis (H0) represents the default assumption that there is no effect or difference, while the alternative hypothesis (H1) suggests that there is a significant effect or difference. For the Chi-Square Goodness-of-Fit Test, the null hypothesis typically states that the observed distribution of preferences matches the expected distribution, while the alternative claims that they do not match.
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Step 1: Write Hypotheses

Significance Level (α)

The significance level, denoted as α, is the threshold for determining 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 support the alternative 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)


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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

In each exercise,

e. interpret the decision in the context of the original claim.

[APPLET] In Exercises 1–3, use the data, which list the hourly wages (in dollars) for randomly selected surgical technologists from three states. Assume the wages are normally distributed and that the samples are independent. (Adapted from U.S. Bureau of Labor Statistics)

Maine: 22.76, 27.60, 25.08, 17.01, 30.15, 27.09, 20.95, 25.52, 20.11, 23.67, 24.32

Oklahoma: 24.64, 21.66, 19.38, 18.19, 23.14, 20.58, 19.53, 30.77, 27.46, 23.80

Massachusetts: 27.07, 24.71, 32.80, 28.34, 33.45, 33.36, 36.81, 30.04, 29.01, 24.30, 29.22, 29.50

Are the mean hourly wages of surgical technologists the same for all three states? Use α=0.01. Assume that the population variances are equal.

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Textbook Question

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)


79
views
Textbook Question

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

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