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)

Verified step by step guidance

1. Step 1: Identify the claim and hypotheses. The claim is that the distribution of people's preferences for payment methods is different from the given distribution. Define the null hypothesis (H₀) as 'The distribution of preferences matches the given distribution' and the alternative hypothesis (Hₐ) as 'The distribution of preferences is different from the given distribution.'
2. Step 2: Calculate the expected frequencies for each payment method based on the given percentages and the total sample size of 600 people. Use the formula: Expected frequency = (Percentage / 100) × Total sample size. For example, for 'Cash,' the expected frequency is (29 / 100) × 600.
3. 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.
4. 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 categories (Cash, Debit or credit, Check, Digital wallet/other), so df = 4 - 1 = 3.
5. Step 5: Compare the calculated χ² value to the critical value from the Chi-Square distribution table at α = 0.01 and df = 3. If the χ² value exceeds the critical value, reject the null hypothesis (H₀). Otherwise, fail to reject H₀.

Key Concepts

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 (H₀ and Hₐ)

In hypothesis testing, the null hypothesis (H₀) represents a statement of no effect or no difference, suggesting that any observed variation is due to chance. The alternative hypothesis (Hₐ) posits that there is a significant effect or difference. In this context, H₀ would state that the distribution of payment preferences matches the expected distribution, while Hₐ would claim that it differs.

Significance Level (α)

The significance level (α) is a threshold set by the researcher to determine the probability of rejecting the null hypothesis when it is actually true, known as a Type I error. In this case, α = 0.01 indicates a 1% risk of concluding that a difference exists when there is none. This level is used to assess the strength of the evidence against the null hypothesis in the context of the Chi-Square test.