To test the hypothesis:

• H₀: μ₁ ≤ μ₂
• H₁: μ₁ > μ₂

We will use the two-sample z-test because we know the population standard deviations.

Steps for conducting the test:

1. Calculate the test statistic: The formula for the z-test for two population means is:

   $z = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$

   Where:

   • $\bar{x}_1 = 2.67$ (sample mean from population 1)
   • $\bar{x}_2 = 2.59$ (sample mean from population 2)
   • $\sigma_1 = 0.75$ (population standard deviation for population 1)
   • $\sigma_2 = 0.66$ (population standard deviation for population 2)
   • $n_1 = 65$ (sample size for population 1)
   • $n_2 = 50$ (sample size for population 2)

2. Substitute the values into the formula:

   $z = \frac{(2.67 - 2.59)}{\sqrt{\frac{0.75^2}{65} + \frac{0.66^2}{50}}}$

3. Calculate the components:

   • The numerator is: $2.67 - 2.59 = 0.08$
   • The denominator: $\sqrt{\frac{0.75^2}{65} + \frac{0.66^2}{50}} = \sqrt{\frac{0.5625}{65} + \frac{0.4356}{50}} = \sqrt{0.0086538 + 0.008712} = \sqrt{0.0173658}$ $\sqrt{0.0173658} \approx 0.1317$

4. Calculate the z-statistic:

   $z = \frac{0.08}{0.1317} \approx 0.607$

5. Find the p-value: Since this is a one-tailed test (because the alternative hypothesis is $H_1: \mu_1 > \mu_2$), we will find the area to the right of $z = 0.607$.

   Using standard z-tables or a z-calculator, the p-value for $z = 0.607$ is approximately 0.271.

Conclusion:

• p-value ≈ 0.271

Since the p-value (0.271) is greater than the significance level (0.08), we fail to reject the null hypothesis.

Would you like further details or have any questions? Here are 5 related questions for you:

1. What is the difference between a one-tailed and a two-tailed hypothesis test?
2. How do you interpret the significance level (α) in hypothesis testing?
3. What assumptions are made when conducting a z-test for two population means?
4. How would the conclusion change if the significance level were 0.10 instead of 0.08?
5. What is the difference between a p-value and the test statistic in hypothesis testing?

Tip: For hypothesis testing, always ensure your data meets the assumptions for the test you're using, especially when dealing with sample sizes and population standard deviations.