The given data is based on the use of hypnotism for pain reduction. The measurements are in centimetreson a pain scale. The measurements are obtained before hypnosis and after hypnosis.

Null Hypothesis: The mean of the differences between the pain measurements before and after hypnosis is equal to 0.

\({H_{0\;}}:\;{\mu _d} = 0\)

Alternative Hypothesis:The mean of the differences between the pain measurements before and after hypnosis is greater than 0.\({H_1}\;:{\mu _d} > 0\;\..

The formula of the confidence interval is as follows:

\({\rm{C}}{\rm{.I}} = \bar d - E < {\mu _d} < \bar d + E\;\)

The value of the margin of error (E) has the following notation:

\(E = {t_{\frac{\alpha }{2},df}} \times \frac{{{s_d}}}{{\sqrt n }}\)

The following table shows the difference in the pain measurements before and after hypnosis:

Now, the mean of the differences between the two pain measurements is computed below:

\(\begin{array}{c}\bar d = \frac{{\sum\limits_{i = 1}^n {{d_i}} }}{n}\\ = \frac{{\left( { - 0.2} \right) + 4.1 + \cdots + 9.6}}{8}\\ = 3.125\end{array}\)

The standard deviation of the differences is equal to:

\(\begin{array}{c}s = \sqrt {\frac{{\sum\limits_{i = 1}^n {{{\left( {{d_i} - \bar d} \right)}^2}} }}{{n - 1}}} \;\\ = \sqrt {\frac{{{{\left( { - 0.2 - 3.125} \right)}^2} + {{\left( {4.1 - 3.125} \right)}^2} + \cdots + {{\left( {9.6 - 3.125} \right)}^2}}}{{8 - 1}}} \\ = 2.911\end{array}\)

The value of the degrees of freedom is equal to:

\(\begin{array}{c}{\rm{df}} = n - 1\\ = 8 - 1\\ = 7\end{array}\)

The confidence level is equal to 95%. Thus, the level of significance becomes 0.05.

Therefore,

\(\begin{array}{c}\frac{\alpha }{2} = \frac{{0.05}}{2}\\ = 0.025\end{array}\)

Referring to the t-distribution table, the critical value of t for 7 degrees of freedom at 0.025 significance level is equal to 2.3646.

The value of the margin of error is computed below:

\(\begin{array}{c}E = {t_{\frac{\alpha }{2},df}} \times \frac{{{s_d}}}{{\sqrt n }}\\ = {t_{\frac{{0.05}}{2},7}} \times \frac{{2.911}}{{\sqrt 8 }}\\ = 2.3646 \times \frac{{2.911}}{{\sqrt 8 }}\\ = 2.433992\end{array}\)

The value of the confidence interval is equal to:

\(\begin{array}{c}\bar d - E < {\mu _d} < \bar d + E\;\\\left( {3.125 - 2.433992} \right) < {\mu _d} < \left( {3.125 + 2.433992} \right)\\0.69 < {\mu _d} < 5.56\end{array}\)

Thus, the 95% confidence interval is equal to (0.69 cm,5.56 cm).

It can be seen that the value of 0 is not included in the interval, and all the values are positive.

This implies that the mean of the differences between the pain measurements cannot be equal to 0.

And, the pain measurement after hypnosis is less than the pain measurement before hypnosis.

Thus, there is enough evidence to conclude that hypnotism appears to be effective in reducing pain.