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The question asks to test whether \(\mu_1 < \mu_2\) at the \(\alpha = 0.02\) level of significance using the provided sample dataThe question asks to test whether \(\mu_1 < \mu_2\) at the \(\alpha = 0.02\) level of significance using the provided sample 

### Data Summary:
- **Population 1**: 
  - \(n_1 = 33\)
  - \(\bar{x}_1 = 103.5\) 
  - \(s_1 = 12.3\)
  
- **Population 2**: 
  - \(n_2 = 25\)
  - \(\bar{x}_2 = 114.5\)
  - \(s_2 = 13.2\)

### Hypotheses:
- \(H_0: \mu_1 = \mu_2\) (Null Hypothesis)
- \(H_1: \mu_1 \neq \mu_2\) (Alternative Hypothesis)

This is a two-tailed test because the alternative hypothesis is a non-equality (\(\mu_1 \neq \mu_2\)).

### Step 1: Calculate the Test Statistic
The formula for the test statistic \( t \) is:

\[
t = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}
\]

Substituting the values:

\[
t = \frac{103.5 - 114.5}{\sqrt{\frac{12.3^2}{33} + \frac{13.2^2}{25}}}
\]

\[
t = \frac{-11}{\sqrt{\frac{151.29}{33} + \frac{174.24}{25}}}
\]

\[
t = \frac{-11}{\sqrt{4.582 + 6.97}} = \frac{-11}{\sqrt{11.552}} = \frac{-11}{3.4} \approx -3.235
\]

### Step 2: Degrees of Freedom (df)
The degrees of freedom for unequal sample sizes and variances can be approximated using the following formula:

\[
df = \frac{\left( \frac{s_1^2}{n_1} + \frac{s_2^2}{n_2} \right)^2}{\frac{\left( \frac{s_1^2}{n_1} \right)^2}{n_1 - 1} + \frac{\left( \frac{s_2^2}{n_2} \right)^2}{n_2 - 1}}
\]

Let me calculate this.The test statistic \( t \) is approximately \(-3.236\), and the degrees of freedom (df) is about 49.8.

### Step 3: Determine the P-value
For a two-tailed test with a \( t \)-value of \(-3.236\) and \( df \approx 50 \), we can find the P-value using a t-distribution table or software.

Let's calculate the P-value.The P-value for this hypothesis test is approximately **0.002**.

### Step 4: Conclusion
Since the P-value (0.002) is less than the significance level \(\alpha = 0.02\), we **reject the null hypothesis**. This means that there is sufficient evidence at the 0.02 level of significance to conclude that \(\mu_1 \neq \mu_2\).

Would you like more details on any part of this calculation?

### 5 Follow-up Questions:
1. How would the test change if the significance level were \(\alpha = 0.05\)?
2. Can you explain how the degrees of freedom formula is derived?
3. What is the impact of sample size on the test statistic?
4. How do we interpret the P-value in a hypothesis test?
5. What is the difference between a one-tailed and two-tailed test?

**Tip:** In hypothesis testing, always compare the P-value with the significance level (\(\alpha\)) to make your decision about rejecting or not rejecting the null hypothesis.

Test whether μ₁ < μ₂ at the α = 0.02 level of significance for the sample data provided. Population 1: n = 33, x̄ = 103.5, s = 12.3. Population 2: n = 25, x̄ = 114.5, s = 13.2.

Learn how to perform a hypothesis test comparing two means at a 0.02 significance level using sample data. The test involves a two-sample t-test and calculates the P-value for populations with unequal variances. Suitable for undergraduate statistics students.

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