Question

Short answer

The 95% confidence interval for the mean of the differences between the pain measurements before hypnosis and after hypnosis is equal to (0.69 cm,5.56 cm).

It seems that there are only positive values in the interval, and the value of 0 does not lie in the interval.

Therefore, theeffect of hypnotism is significant in reducing pain.

Step-by-step solution

Given information

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.

Hypothesis for problem

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\;\..

Expression of the confidence interval

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:

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

Table of the differences

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

Subject	A	B	C	D	E	F	G	H
Before	6.6	6.5	9	10.3	11.3	8.1	6.3	11.6
After	6.8	2.4	7.4	8.5	8.1	6.1	3.4	2
Differences (d)	-0.2	4.1	1.6	1.8	3.2	2	2.9	9.6

Find the mean of the differences

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

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

Find the standard deviation of the differences

The standard deviation of the differences is equal to:

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

Find the critical value

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.

Compute the margin of error

The value of the margin of error is computed below:

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

Step 9:Computethe confidence interval

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).

Interpretation of the confidence interval

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.