In Exercises 13–18, test the claim about the difference between two population variances σ₁² and σ₂² at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.

Claim: σ₁² ≤ σ₂²; α = 0.01.
Sample statistics: s₁² = 842, n₁ = 11 and s₂² = 836, n₂ = 10

검증된 단계별 안내

Step 1: Identify the null hypothesis (H₀) and the alternative hypothesis (H₁). The claim is that σ₁² ≤ σ₂². Therefore, the null hypothesis is H₀: σ₁² ≤ σ₂², and the alternative hypothesis is H₁: σ₁² > σ₂². This is a one-tailed test.

Step 2: Calculate the test statistic using the F-distribution formula: F = (s₁² / s₂²), where s₁² and s₂² are the sample variances. Substitute the given values: s₁² = 842 and s₂² = 836.

Step 3: Determine the degrees of freedom for the numerator (df₁) and denominator (df₂). For the numerator, df₁ = n₁ - 1, and for the denominator, df₂ = n₂ - 1. Substitute the sample sizes n₁ = 11 and n₂ = 10 to calculate df₁ and df₂.

Step 4: Find the critical value of F from the F-distribution table at the significance level α = 0.01 for a one-tailed test, using the calculated degrees of freedom (df₁ and df₂).

Step 5: Compare the calculated F value to the critical F value. If the calculated F value is greater than the critical F value, reject the null hypothesis H₀. Otherwise, fail to reject H₀. Interpret the result in the context of the claim.

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주요 개념

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Hypothesis Testing

Hypothesis testing is a statistical method used to make decisions about population parameters based on sample data. It involves formulating a null hypothesis (H0) and an alternative hypothesis (H1), then using sample statistics to determine whether to reject H0. In this case, the null hypothesis states that the variance of the first population is less than or equal to that of the second (σ₁² ≤ σ₂²).

F-Test for Variances

The F-test is a statistical test used to compare two population variances. It calculates the ratio of the two sample variances (s₁²/s₂²) and compares it to a critical value from the F-distribution based on the degrees of freedom of the samples. This test helps determine if there is a significant difference between the variances, which is essential for validating the claim in the question.

Level of Significance (α)

The level of significance, denoted as α, is the probability of rejecting the null hypothesis when it is actually true (Type I error). In this scenario, α is set at 0.01, indicating a 1% risk of concluding that a difference exists when there is none. This threshold helps determine the critical value for the F-test and influences the strength of evidence required to support the claim.