Math Problem Statement
Solution
The image shows a hypothesis testing problem based on independent random samples from two populations with unequal variances. Let's break it down.
Given:
- Two samples of data are provided for Population 1 (Sample 1) and Population 2 (Sample 2).
- You are tasked with determining whether there is sufficient evidence to conclude that (i.e., whether the means of the two populations are different).
- The significance level .
- There are also three hypothesis options to select from (A, B, or C).
Step 1: Define the Hypotheses
The problem asks us to test whether the means of the two populations are different. The null and alternative hypotheses are based on this:
- Null Hypothesis : (the population means are equal)
- Alternative Hypothesis : (the population means are not equal)
This corresponds to a two-tailed test because the alternative hypothesis suggests the possibility that could be either greater than or less than .
Thus, the correct hypotheses should be:
This corresponds to option A.
Step 2: Hypothesis Testing Procedure
You would likely conduct a two-sample t-test to compare the means of the two populations because:
- The samples are independent.
- The population variances are unequal (which suggests we should use the Welch’s t-test).
The test would involve the following steps:
- Calculate the sample means and for Sample 1 and Sample 2, respectively.
- Estimate the standard errors based on the sample variances and sizes.
- Calculate the test statistic (t-value) and compare it with the critical value based on the t-distribution and .
- Make a decision based on whether the test statistic falls in the rejection region (i.e., whether we reject or fail to reject ).
Would you like further details on how to compute the t-test, or do you have any specific questions on this?
Follow-up questions:
- What are the sample means and standard deviations for each sample?
- How is Welch’s t-test different from the standard two-sample t-test?
- How do you calculate the degrees of freedom in Welch’s t-test?
- What is the critical t-value for a two-tailed test at ?
- How would the decision change if were 0.05 instead of 0.01?
Tip:
When working with unequal variances, always verify if Welch's t-test is appropriate instead of the standard pooled t-test.
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Math Problem Analysis
Mathematical Concepts
Hypothesis Testing
Two-Sample t-Test
Significance Level
Welch's t-Test
Formulas
Null Hypothesis: H0: μ1 = μ2
Alternative Hypothesis: H1: μ1 ≠ μ2
Test Statistic: t = (x̄1 - x̄2) / √(s1^2/n1 + s2^2/n2)
Theorems
Welch's t-Test
Suitable Grade Level
College or University level
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