Step 1 :Calculate the mean of the differences: \( \bar{d} = \frac{\sum d_i}{n} = \frac{0.18 + 0.092 - 0.084 + 0.207 + 0.183 - 0.227 - 0.022 + 0.132 - 0.087 + 0.013 + 0.076 - 0.015 + 0.081}{13} = 0.040692307692307694 \)
Step 2 :Calculate the standard deviation of the differences: \( s_d = \sqrt{\frac{\sum (d_i - \bar{d})^2}{n-1}} \)
Step 3 :Use the t-distribution to find the critical value for a 95% confidence level with 12 degrees of freedom: \( t_{\text{critical}} = 2.1788128296634177 \)
Step 4 :Calculate the margin of error: \( E = t_{\text{critical}} \times \frac{s_d}{\sqrt{n}} = 2.1788128296634177 \times \frac{0.12664542669423204}{\sqrt{13}} = 0.07653106540943064 \)
Step 5 :Calculate the lower limit of the confidence interval: \( \bar{d} - E = 0.040692307692307694 - 0.07653106540943064 = -0.03583875771712294 \)
Step 6 :Calculate the upper limit of the confidence interval: \( \bar{d} + E = 0.040692307692307694 + 0.07653106540943064 = 0.11722337310173833 \)
Step 7 :Round the final answers to three decimal places: Lower limit: \( \boxed{-0.036} \), Upper limit: \( \boxed{0.117} \)