Performing a Chi-Square Goodness-of-Fit Test
In Exercises 7–16, (e) interpret the decision in the context of the original claim.

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) assumes that the distribution of people's preferences matches the given distribution (29% for cash, 59% for debit/credit, 5% for check, and 7% for digital wallet/other). The alternative hypothesis (H₁) claims that the distribution of people's preferences is different from the given distribution.

Step 2: Calculate the expected frequencies for each category. Multiply the total sample size (600) by the proportions given in the figure. For example, the expected frequency for cash is 600 × 0.29, for debit/credit is 600 × 0.59, and so on.

Step 3: Compute the chi-square test statistic using the formula: χ² = Σ((Oᵢ - Eᵢ)² / Eᵢ), where Oᵢ represents the observed frequency and Eᵢ represents the expected frequency for each category. Perform this calculation for each category (cash, debit/credit, check, digital wallet/other) and sum the results.

Step 4: Determine the critical value for the chi-square test. Use the chi-square distribution table with degrees of freedom (df = number of categories - 1) and the significance level α = 0.01. In this case, df = 4 - 1 = 3.

Step 5: Compare the calculated chi-square test statistic to the critical value. If the test statistic exceeds the critical value, reject the null hypothesis. Otherwise, fail to reject the null hypothesis. Interpret the decision in the context of the original claim about people's payment preferences.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 based on a specific hypothesis. It compares the actual data collected from a sample to a theoretical distribution, allowing researchers to assess whether the sample data fits the expected distribution.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) represents the default position 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.

Significance Level (α)

The significance level, denoted as α, is the threshold used 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 support the alternative hypothesis.