10. Hypothesis Testing for Two Samples

Two Means - Unknown, Unequal Variance

10. Hypothesis Testing for Two Samples

Two Means - Unknown, Unequal Variance

1:29 minutes

Problem 8.1.25

Textbook Question

Getting at the Concept Explain why the null hypothesis Ho: μ1=μ2 is equivalent to the null hypothesis .Ho: μ1-μ2=0

Verified step by step guidance

Understand the null hypothesis Ho: μ1 = μ2. This hypothesis states that the population means of two groups (μ1 and μ2) are equal, implying no difference between the two groups.

Recognize that the null hypothesis Ho: μ1 - μ2 = 0 is another way of expressing the same idea. If the difference between the two population means (μ1 - μ2) is zero, it means the two means are equal.

Mathematically, the equivalence can be shown by rearranging the equation μ1 = μ2. Subtract μ2 from both sides to get μ1 - μ2 = 0. This demonstrates that the two forms of the null hypothesis are interchangeable.

Conceptually, both hypotheses test whether there is no difference between the two population means. The choice of expression depends on the context or the statistical test being used, but they represent the same null hypothesis.

In practice, statistical tests like the t-test often use the form Ho: μ1 - μ2 = 0 because it directly relates to the calculation of the test statistic, which is based on the difference between sample means.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Null Hypothesis

The null hypothesis is a fundamental concept in statistics that posits no effect or no difference between groups or conditions. It serves as a default position that indicates any observed effect in data is due to sampling variability. In hypothesis testing, the null hypothesis is typically denoted as H0 and is tested against an alternative hypothesis (H1) that suggests a significant effect or difference exists.

Population Means

Population means refer to the average values of a particular characteristic within a defined group. In the context of the null hypothesis H0: μ1=μ2, it asserts that the means of two populations (μ1 and μ2) are equal. This concept is crucial for comparing groups in statistical tests, as it helps determine whether any observed differences in sample means are statistically significant or not.

Difference of Means

The difference of means is a statistical measure that quantifies the disparity between the average values of two groups. The expression H0: μ1-μ2=0 is an alternative way to represent the null hypothesis, indicating that the difference between the two population means is zero. This formulation is often used in hypothesis testing to assess whether the observed difference in sample means is significant enough to reject the null hypothesis.

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Textbook Question

In Exercises 11–14, test the claim about the difference between two population means and at the level of significance . Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1<μ2; α=0.03

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Textbook Question

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Textbook Question

Test the claim about the difference between two population means and at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.

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Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

What conditions are necessary to use the t-test for testing the difference between two population means?

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