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.