91Ó°ÊÓ

\(= 0.8\) #tag_title#Step 2: Calculate the standard error for the difference in means#tag_content# Next, compute the standard error for the difference in means using the formula: $$SE(\underline{X} - \underline{Y}) = \sqrt{\frac{S_{1}^{2}}{n} + \frac{S_{2}^{2}}{m}} = \sqrt{\frac{49}{10} + \frac{32}{7}}\) \(= \sqrt{7.142}\) #tag_title#Step 3: Find the t-value for the given confidence level#tag_content# Now, find the t-value corresponding to a 90% confidence level using the degrees of freedom: $$df = min(n - 1, m - 1) = min(9, 6) = 6$$ Consulting a t-table or using calculator yields \(t = 1.943\). #tag_title#Step 4: Calculate the margin of error#tag_content# Multiply the standard error by the t-value to find the margin of error: $$ME = t \times SE = 1.943 \times \sqrt{7.142} = 4.16\) #tag_title#Step 5: Construct the confidence interval#tag_content# Finally, construct the 90% confidence interval for the difference in means: $$(\underline{X} - \underline{Y}) \pm ME = 0.8 \pm 4.16$$ The 90% confidence interval for \(\mu_{1} - \mu_{2}\) is \((-3.36, 4.96)\).