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?
Before
After
8.8
6.2
4.1
2.4
8.4
7.2
11.7
8.3
12.5
8.3
5.9
6.4
5.9
3.6
1.5
2.9
Construct a $95 \%$ confidence interval for the mean of the "before - after" differences.
$0.11< \mu_{d}< 3.27$ (Round to two decimal places as needed.)
Does hypnotism appear to be effective in reducing pain?
A. No, because the confidence interval includes zero.
B. No, because the confidence interval does not include zero and is entirely greater than zero.
C. Yes, because the confidence interval includes zero.
D. Yes, because the confidence interval does not include zero and is entirely greater than zero.

Step 1 ：Given the 'before' and 'after' measurements, we first calculate the differences between these two sets of measurements. The differences are calculated as 'before - after'.

Step 2 ：The differences are: \(2.6, 1.7, 1.2, 3.4, 4.2, -0.5, 2.3, -1.4\).

Step 3 ：We then calculate the mean and standard deviation of these differences. The mean difference is approximately \(1.69\) and the standard deviation is approximately \(1.89\).

Step 4 ：We use the formula for a confidence interval, which is given by \(\bar{x} \pm z \frac{s}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(z\) is the z-score corresponding to the desired confidence level (for a 95% confidence level, \(z = 1.96\)), \(s\) is the sample standard deviation, and \(n\) is the sample size.

Step 5 ：Substituting the values into the formula, we get the 95% confidence interval for the mean of the 'before - after' differences as approximately \((0.38, 3.00)\).

Step 6 ：Since this interval does not include zero and is entirely greater than zero, it indicates that there is a significant difference in pain before and after hypnotism. Therefore, hypnotism appears to be effective in reducing pain.

Step 7 ：Final Answer: \(\boxed{\text{(D) Yes, because the confidence interval does not include zero and is entirely greater than zero.}}\)