In Exercises 5–8, 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

Population statistics: σ1= 0.11 and σ2= 0.10

Sample statistics: x̅1 = 0.28, n1 = 41, and x̅2= 0.33, n2= 34

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Identify the null and alternative hypotheses based on the claim μ1 < μ2. The null hypothesis (H0) is μ1 ≥ μ2, and the alternative hypothesis (H1) is μ1 < μ2.

Determine the significance level α = 0.10, which will be used to find the critical value for the test.

Since the population standard deviations σ1 and σ2 are known, use the z-test for the difference between two means. Calculate the test statistic using the formula: \[\text{z} = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}\] Note that under the null hypothesis, (μ1 - μ2) = 0.

Calculate the standard error of the difference between the sample means using: \[\text{SE} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}\]

Compare the calculated z-test statistic to the critical z-value for a left-tailed test at α = 0.10. If the test statistic is less than the critical value, reject the null hypothesis; otherwise, do not reject it.

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주요 개념

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Hypothesis Testing for Two Population Means

This involves formulating null and alternative hypotheses about the difference between two population means (μ1 and μ2). The goal is to determine if there is enough evidence to support the claim (μ1 < μ2) using sample data, by comparing a test statistic to a critical value or p-value at a given significance level.

Sampling Distribution and Test Statistic

The test statistic measures how far the observed difference between sample means (x̅1 - x̅2) is from the hypothesized difference under the null hypothesis. When population standard deviations (σ1, σ2) are known and samples are independent and normal, the test statistic follows a normal distribution, allowing calculation of z-scores.

Significance Level and Decision Rule

The significance level (α = 0.10) defines the probability of rejecting the null hypothesis when it is true (Type I error). It sets the critical value(s) for the test statistic. If the test statistic falls into the rejection region determined by α, the null hypothesis is rejected in favor of the alternative claim.