In Exercises 11–16, test the claim about the difference between two population means μ1 and μ2 at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1> μ2; α=0.10. Assume (σ1)^2 ≠ (σ2)^2

Sample statistics: x̅1= 520, s1= 25, n1= 7 and x̅2= 500, s2= 55, n2= 6

검증된 단계별 안내

Step 1: Identify the null and alternative hypotheses. The null hypothesis (H₀) is μ₁ ≤ μ₂, and the alternative hypothesis (H₁) is μ₁ > μ₂. This is a one-tailed test since the claim specifies μ₁ > μ₂.

Step 2: Determine the test statistic formula for comparing two population means when the population variances are unequal. Use the formula:

\frac{({x̅}_{1} - {x̅}_{2})}{\sqrt{s_{1}^{2} + s_{2}^{2}}}

, where x̅₁ and x̅₂ are the sample means, s₁² and s₂² are the sample variances, and n₁ and n₂ are the sample sizes.

Step 3: Calculate the degrees of freedom using the formula for unequal variances:

\frac{(s_{1}^{2} + s_{2}^{2}) 2}{s_{1}^{4} + s_{2}^{4}}

. This will give an approximate degrees of freedom for the t-distribution.

Step 4: Use the calculated test statistic and degrees of freedom to find the critical value from the t-distribution table at the significance level α = 0.10 for a one-tailed test. Compare the test statistic to the critical value to determine whether to reject or fail to reject the null hypothesis.

Step 5: Interpret the results. If the test statistic exceeds the critical value, reject the null hypothesis and conclude that there is sufficient evidence to support the claim μ₁ > μ₂. Otherwise, fail to reject the null hypothesis and conclude that there is insufficient evidence to support 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 favor of H1. In this case, the claim is that the mean of population 1 (μ1) is greater than the mean of population 2 (μ2).

Level of Significance (α)

The level of significance, denoted as α, is the probability of rejecting the null hypothesis when it is actually true. It represents the threshold for determining whether the observed data is statistically significant. In this question, α is set at 0.10, meaning there is a 10% risk of concluding that μ1 is greater than μ2 when it is not.

Independent Samples and Variance

Independent samples refer to two groups that are not related or paired in any way, allowing for the comparison of their means. In this scenario, the assumption that the variances of the two populations are not equal (σ1^2 ≠ σ2^2) indicates the use of a specific statistical test, such as Welch's t-test, which adjusts for the difference in variances when comparing the means.