To construct a confidence interval for the difference between two population means $\mu_1$ and $\mu_2$ with known standard deviations, we use the following formula for the confidence interval of the difference in means:

$\left( \bar{x}_1 - \bar{x}_2 \right) \pm z_{\alpha/2} \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}$

Step-by-Step Solution

Given Information

• Hypotheses:
  • Null hypothesis ($H_0$): $\mu_1 = \mu_2$
  • Alternative hypothesis ($H_1$): $\mu_1 \neq \mu_2$
• Known standard deviations:
  • $\sigma_1 = 10$
  • $\sigma_2 = 5$
• Sample sizes:
  • $n_1 = 10$
  • $n_2 = 15$
• Sample means:
  • $\bar{x}_1 = 4.7$
  • $\bar{x}_2 = 7.8$
• Significance level:
  • $\alpha = 0.05$

Since this is a two-tailed test (due to $H_1: \mu_1 \neq \mu_2$), we will use $z_{\alpha/2} = z_{0.025}$. For $\alpha = 0.05$, $z_{0.025} \approx 1.96$.

Step 1: Calculate the Difference in Sample Means

$\bar{x}_1 - \bar{x}_2 = 4.7 - 7.8 = -3.1$

Step 2: Calculate the Standard Error of the Difference

$\text{Standard Error} = \sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}} = \sqrt{\frac{10^2}{10} + \frac{5^2}{15}}$ $= \sqrt{\frac{100}{10} + \frac{25}{15}} = \sqrt{10 + 1.6667} = \sqrt{11.6667} \approx 3.42$

Step 3: Calculate the Margin of Error

$\text{Margin of Error} = z_{\alpha/2} \times \text{Standard Error} = 1.96 \times 3.42 \approx 6.7$

Step 4: Construct the Confidence Interval

$\text{Confidence Interval} = \left( \bar{x}_1 - \bar{x}_2 \right) \pm \text{Margin of Error}$ $= -3.1 \pm 6.7$ $= (-9.8, 3.6)$

Interpretation of the Confidence Interval

The 95% confidence interval for the difference in population means $\mu_1 - \mu_2$ is $(-9.8, 3.6)$.

How the Test Could be Conducted with a Confidence Interval

In a hypothesis test, we reject the null hypothesis $H_0$ if the confidence interval for $\mu_1 - \mu_2$ does not contain 0. Here, the interval $(-9.8, 3.6)$ includes 0, so we do not reject the null hypothesis at the $\alpha = 0.05$ significance level. This suggests that there is insufficient evidence to conclude that $\mu_1$ and $\mu_2$ are different.

Would you like further clarification on this topic?

Here are some related questions you might find useful:

1. How does the choice of confidence level affect the width of the confidence interval?
2. What changes in the confidence interval calculation if we assume unequal variances?
3. How would a one-tailed test alter the confidence interval interpretation?
4. How can we conduct a similar test if the sample sizes are much smaller?
5. What are Type I and Type II errors, and how do they relate to this test?

Tip: Remember that a confidence interval provides an estimate for the range in which the true population parameter lies, with a given level of certainty (95% in this case). It can also inform hypothesis testing by indicating whether a hypothesized value (like 0) is plausible.