Exercises 1–5 refer to the sample data in the following table, which summarizes the frequencies of 500 digits randomly generated by Statdisk. Assume that we want to use a 0.05 significance level to test the claim that Statdisk generates the digits in a way that they are equally likely.





What are the null and alternative hypotheses corresponding to the stated claim?

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to test the claim that Statdisk generates digits in a way that they are equally likely. This involves setting up hypotheses for a chi-square goodness-of-fit test.
2. Step 2: Define the null hypothesis (H₀). The null hypothesis states that the digits are equally likely to occur, meaning the expected frequency for each digit is the same. Mathematically, H₀: P(0) = P(1) = P(2) = ... = P(9).
3. Step 3: Define the alternative hypothesis (H₁). The alternative hypothesis states that the digits are not equally likely to occur, meaning at least one digit has a different probability. Mathematically, H₁: At least one P(i) ≠ P(j) for i ≠ j.
4. Step 4: Calculate the expected frequency for each digit under the null hypothesis. Since there are 500 total digits and 10 possible digits, the expected frequency for each digit is 500 ÷ 10 = 50.
5. Step 5: Use the observed frequencies from the table and the expected frequencies to compute the chi-square test statistic. The formula for the chi-square test statistic is χ² = Σ((Oᵢ - Eᵢ)² / Eᵢ), where Oᵢ is the observed frequency and Eᵢ is the expected frequency for each digit.

Key Concepts

Null Hypothesis (H0)

The null hypothesis is a statement that assumes no effect or no difference, serving as a default position in hypothesis testing. In this context, it posits that the digits generated by Statdisk are equally likely, meaning each digit from 0 to 9 has the same probability of occurrence. This hypothesis is tested against the alternative hypothesis to determine if there is enough evidence to reject it.

Alternative Hypothesis (H1)

The alternative hypothesis is a statement that contradicts the null hypothesis, suggesting that there is an effect or a difference. For this scenario, the alternative hypothesis would claim that the digits generated by Statdisk are not equally likely, indicating that some digits occur more frequently than others. This hypothesis is what researchers aim to support through statistical testing.

Significance Level (α)

The significance level, denoted as alpha (α), is the threshold for determining whether to reject the null hypothesis. In this case, a significance level of 0.05 indicates that there is a 5% risk of concluding that a difference exists when there is none. If the p-value obtained from the statistical test is less than 0.05, the null hypothesis would be rejected in favor of the alternative hypothesis.