A researcher is conducting a hypothesis test with the following information:

H
0
:
μ
2
−
μ
1
=
0

H
1
:
μ
2
−
μ
1
≠
0

¯
x
1
= 50.4
s
1
= 2.14
n
1
= 11

¯
x
2
= 47.5
s
2
= 2.36
n
2
= 13

α
= 0.01

What type of test is being conducted?
a right-tailed test
a left-tailed test
a two-tailed test

What critical value would be the cutoff for the rejection region? Round your answer to three decimal places. For a two-tailed test, enter only the positive critical value.

What is the test statistic for the test based on the sample data? Round your answer to three decimal places.

What decision results from this test?
reject the null hypothesis
fail to reject the alternate hypothesis
support the null hypothesis
fail to reject the null hypothesis

Let's analyze the hypothesis test step by step.

### 1. **Type of Test**
The hypotheses are given as:
- Null Hypothesis (\(H_0\)): \(\mu_2 - \mu_1 = 0\)
- Alternative Hypothesis (\(H_1\)): \(\mu_2 - \mu_1 \neq 0\)

Since the alternative hypothesis is looking for any difference (either positive or negative) between the two means, this is a **two-tailed test**.

### 2. **Critical Value**
For a two-tailed test with a significance level \(\alpha = 0.01\), the critical values correspond to the area in both tails of the normal distribution. The critical value for a two-tailed test at \(\alpha = 0.01\) can be found using the standard normal distribution (Z-distribution) table.

At \(\alpha = 0.01\), the area in each tail is \(\alpha/2 = 0.005\). The Z-score corresponding to 0.005 in the tail is approximately **2.576**. 

So, the positive critical value is:
\[
\text{Critical value} = 2.576
\]

### 3. **Test Statistic**
To calculate the test statistic, we use the formula for the two-sample Z-test for means:

\[
Z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}
\]

Given:
\[
\bar{x}_1 = 50.4, \quad s_1 = 2.14, \quad n_1 = 11
\]
\[
\bar{x}_2 = 47.5, \quad s_2 = 2.36, \quad n_2 = 13
\]
\[
\mu_1 - \mu_2 = 0
\]

Substituting these values:

\[
Z = \frac{(50.4 - 47.5) - 0}{\sqrt{\frac{2.14^2}{11} + \frac{2.36^2}{13}}}
\]

Now, calculate the value step by step:

\[
Z = \frac{2.9}{\sqrt{\frac{4.5796}{11} + \frac{5.5696}{13}}}
\]
\[
Z = \frac{2.9}{\sqrt{0.416327 + 0.4284308}} = \frac{2.9}{\sqrt{0.8447578}} = \frac{2.9}{0.919144} \approx 3.155
\]

So, the test statistic \(Z\) is approximately **3.155**.

### 4. **Decision**
The decision rule is:
- Reject the null hypothesis if \(|Z| > 2.576\) (since this is a two-tailed test).

Since the calculated test statistic \(Z = 3.155\) is greater than the critical value \(2.576\), we **reject the null hypothesis**.

### Summary:
- **Type of test:** Two-tailed test
- **Critical value:** 2.576
- **Test statistic:** 3.155
- **Decision:** Reject the null hypothesis

Would you like further details or have any questions about the steps involved?

Here are 8 related questions:
1. What is the difference between a one-tailed and a two-tailed test?
2. How is the critical value determined for different significance levels?
3. What assumptions are required for the two-sample Z-test to be valid?
4. How would the test statistic change if the sample sizes were larger?
5. What is the relationship between the p-value and the test statistic?
6. How does the choice of \(\alpha\) affect the outcome of the hypothesis test?
7. What are Type I and Type II errors in the context of hypothesis testing?
8. How would you interpret the test results if the null hypothesis was not rejected?

**Tip:** Always check if the conditions for the hypothesis test are met, such as normality of data and equal variances, before proceeding with the test.

Explore how to conduct a two-sample Z-test for means using a detailed example. Understand the hypotheses, calculate critical values, and interpret test statistics with sample data. Ideal for advanced undergraduate students and researchers.

Math Problem Statement

H 0 : μ 2 − μ 1

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