A study was conducted to measure the effectiveness of hypnotism in reducing pain. The measurements are centimeters on a pain scale before and after hypnosis. Assume that the paired sample data are simple random samples and that the differences have a distribution that is approximately normal. Construct a 95\% confidence interval for the mean of the "before-after" differences. Does hypnotism appear to be effective in reducing pain?
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Construct a $95\mathrm{\%}$ confidence interval for the mean of the "before-after" differences.
$<{\mu }_{\mathrm{d}}<◻$ (Round to two decimal places as needed.)
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Step 1 ：First, calculate the differences between the 'before' and 'after' measurements for each individual. The differences are $[3.9,0.1,0.7,0.9,-0.6,2.6,1.3,1.6]$.

Step 2 ：Next, calculate the mean and standard deviation of these differences. The mean difference is approximately $1.31$ and the standard deviation is approximately $1.42$.

Step 3 ：Then, calculate the 95% confidence interval for the mean difference using the formula: mean ± (1.96 * (standard deviation / sqrt(n))), where n is the number of pairs. In this case, n is 8.

Step 4 ：The lower limit of the confidence interval is $0.33$ and the upper limit is $2.30$.

Step 5 ：Final Answer: The 95% confidence interval for the mean of the 'before-after' differences is $\boxed{{\displaystyle [0.33,2.30]}}$. This means we are 95% confident that the true mean difference in pain levels before and after hypnosis is between 0.33 and 2.30 centimeters on the pain scale. Since the entire interval is above zero, this suggests that hypnotism is effective in reducing pain.