Math Problem Statement

Do the data present sufficient evidence to conclude that μ1 ≠ μ2? Use α = 0.01. Let μ1 be the mean of population 1 and μ2 be the mean of population 2. Select the correct hypotheses.

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 μ1μ2\mu_1 \neq \mu_2 (i.e., whether the means of the two populations are different).
  • The significance level α=0.01\alpha = 0.01.
  • 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 H0H_0: μ1=μ2\mu_1 = \mu_2 (the population means are equal)
  • Alternative Hypothesis H1H_1: μ1μ2\mu_1 \neq \mu_2 (the population means are not equal)

This corresponds to a two-tailed test because the alternative hypothesis suggests the possibility that μ1\mu_1 could be either greater than or less than μ2\mu_2.

Thus, the correct hypotheses should be:

  • H0:μ1=μ2H_0: \mu_1 = \mu_2
  • H1:μ1μ2H_1: \mu_1 \neq \mu_2

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:

  1. Calculate the sample means x1ˉ\bar{x_1} and x2ˉ\bar{x_2} for Sample 1 and Sample 2, respectively.
  2. Estimate the standard errors based on the sample variances and sizes.
  3. Calculate the test statistic (t-value) and compare it with the critical value based on the t-distribution and α=0.01\alpha = 0.01.
  4. Make a decision based on whether the test statistic falls in the rejection region (i.e., whether we reject or fail to reject H0H_0).

Would you like further details on how to compute the t-test, or do you have any specific questions on this?

Follow-up questions:

  1. What are the sample means and standard deviations for each sample?
  2. How is Welch’s t-test different from the standard two-sample t-test?
  3. How do you calculate the degrees of freedom in Welch’s t-test?
  4. What is the critical t-value for a two-tailed test at α=0.01\alpha = 0.01?
  5. How would the decision change if α\alpha 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.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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