Explain how to perform a​ two-sample t-test for the difference between two population means. 
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Part 1
Which explanation below best describes how to perform the​ two-sample t-test?
A.
State the hypotheses and identify the claim. Specify the level of significance. Determine the degrees of freedom. Find the critical​ value(s) and identify the rejection​ region(s). Find the standardized test statistic. Make a decision and interpret it in the context of the original claim. 
B.
Find the standardized test statistic. State the hypotheses and identify the claim. Specify the level of significance. Determine the degrees of freedom. Find the critical​ value(s) and identify the rejection​ region(s). Make a decision and interpret it in the context of the original claim. 
C.
Determine the degrees of freedom. Specify the level of significance. State the hypotheses and identify the claim. Find the critical​ value(s) and identify the rejection​ region(s). Find the standardized test statistic. Make a decision and interpret it in the context of the original claim. 
D.
Find the critical​ value(s) and identify the rejection​ region(s). Specify the level of significance. State the hypotheses and identify the claim. Determine the degrees of freedom. Make a decision and interpret it in the context of the original claim. Find the standardized test statistic.

The correct explanation for performing a two-sample t-test for the difference between two population means is best described by option A. Here is a detailed explanation:

1. **State the hypotheses and identify the claim**: Formulate the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)). The null hypothesis typically states that there is no difference between the population means, while the alternative hypothesis states that there is a difference.

2. **Specify the level of significance**: Choose the significance level (\(\alpha\)), which is the probability of rejecting the null hypothesis when it is actually true. Common choices are 0.05, 0.01, or 0.10.

3. **Determine the degrees of freedom**: The degrees of freedom for a two-sample t-test can be calculated using a specific formula that takes into account the sample sizes and variances of both samples. The formula is:
\[ 
df = \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{\left( \frac{s_1^2}{n_1} \right)^2}{n_1 - 1} + \frac{\left( \frac{s_2^2}{n_2} \right)^2}{n_2 - 1}} 
\]
where \(s_1^2\) and \(s_2^2\) are the sample variances, and \(n_1\) and \(n_2\) are the sample sizes.

4. **Find the critical value(s) and identify the rejection region(s)**: Determine the critical t-value(s) from the t-distribution table using the specified significance level and degrees of freedom. Define the rejection region(s) based on these critical values.

5. **Find the standardized test statistic**: Calculate the t-statistic using the formula:
\[ 
t = \frac{(\bar{X}_1 - \bar{X}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}} 
\]
where \(\bar{X}_1\) and \(\bar{X}_2\) are the sample means, and \(\mu_1\) and \(\mu_2\) are the population means under the null hypothesis (usually \(\mu_1 - \mu_2 = 0\)).

6. **Make a decision**: Compare the calculated t-statistic with the critical value(s). If the t-statistic falls into the rejection region, reject the null hypothesis. Otherwise, fail to reject the null hypothesis.

7. **Interpret the decision in the context of the original claim**: Explain what the statistical decision means in terms of the real-world context of the problem.

Given the sequence of steps outlined above, the best description is:

**A. State the hypotheses and identify the claim. Specify the level of significance. Determine the degrees of freedom. Find the critical value(s) and identify the rejection region(s). Find the standardized test statistic. Make a decision and interpret it in the context of the original claim.**

Would you like more details on any of these steps or have any questions?

Here are some related questions you might find helpful:

1. How do you calculate the pooled variance for a two-sample t-test?
2. What assumptions must be met to perform a two-sample t-test?
3. How do you interpret the p-value in a two-sample t-test?
4. Can you explain the difference between a one-tailed and a two-tailed t-test?
5. What is the effect of sample size on the power of a two-sample t-test?
6. How do you handle unequal variances in a two-sample t-test?
7. What are the consequences of violating the assumptions of a two-sample t-test?
8. How does the choice of significance level (\(\alpha\)) affect the results of a two-sample t-test?

**Tip:** Always check the assumptions of normality and equal variances before performing a two-sample t-test to ensure valid results.

Learn how to perform a two-sample t-test for the difference between two population means using detailed step-by-step instructions. Includes hypothesis testing, significance level determination, degrees of freedom calculation, critical values identification, standardized test statistic computation, decision making, and interpretation. Ideal for advanced learners and researchers.

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