91Ó°ÊÓ

A study was conducted to investigate the effectiveness of hypnotism in reducing pain. Results for randomly selected subjects are given in the accompanying table (based on "An Analysis of Factors That Contribute to the Efficacy of Hypnotic Analgesia," by Price and Barber, Journal of Abnormal Psychology, Vol. 96, No. 1 ). The values are before and after hypnosis; the measurements are in centimeters on a pain scale. Higher values correspond to greater levels of pain. Construct a \(95 \%\) confidence interval for the mean of the "before/after" differences. Does hypnotism appear to be effective in reducing pain? $$ \begin{array}{l|c|c|c|c|c|c|c|c|} \hline \text { Subject } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } & \text { F } & \text { G } & \text { H } \\ \hline \text { Before } & 6.6 & 6.5 & 9.0 & 10.3 & 11.3 & 8.1 & 6.3 & 11.6 \\ \hline \text { After } & 6.8 & 2.4 & 7.4 & 8.5 & 8.1 & 6.1 & 3.4 & 2.0 \\ \hline \end{array} $$

Step by step solution

01

Calculate the differences

For each subject, calculate the difference between the 'before' and 'after' pain measurements. For example, for Subject A: Difference = 6.6 - 6.8 = -0.2.

02

Compute the mean difference

Sum all the differences from Step 1 and then divide by the number of subjects to find the mean difference, \(\bar{d}\).

03

Compute the standard deviation of differences

Calculate the standard deviation of the differences found in Step 1 using the formula: \[s_d = \sqrt{\frac{\sum (d_i - \bar{d})^2}{n-1}}\], where \({d_i}\) are the individual differences and \(n\) is the number of subjects.

04

Determine the standard error of the mean difference

Calculate the standard error of the mean difference, SE\(\bar{d}\) using: \[SE_{\bar{d}} = \frac{s_d}{\sqrt{n}}\].

05

Find the t critical value

Use a t-table to find the critical t-value (t_c) for a 95% confidence interval with \(n-1\) degrees of freedom.

06

Construct the confidence interval

Construct the \(95\%\) confidence interval for the mean difference using: \[CI = \bar{d} \pm (t_c \cdot SE_{\bar{d}})\].

07

Interpret the results

Analyze if the confidence interval includes zero. If it does not, it suggests that hypnotism may be effective in reducing pain.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

hypnotism and pain reduction

Hypnotism is often explored as a method for relieving pain without the need for medication. In the context of this study, pain levels were measured before and after hypnosis to determine its effectiveness. Higher numbers on the pain scale mean more intense pain. A lower number after hypnosis implies that the technique might be working to reduce pain. By examining these changes in pain levels, researchers aim to figure out if hypnotism can statistically be proven to alleviate discomfort.

95% confidence interval

A 95% confidence interval provides a range within which we can be 95% sure that the true mean difference lies. This is based on our sample data. Here, we calculated it for the differences in pain levels before and after hypnosis. To construct this interval:

• First, we find the mean of the differences.
• Second, we calculate the standard deviation of those differences.
• Next, we use these to find the standard error.
• We then determine the appropriate t-value from the t-distribution table for our sample size.
• Finally, we use these values to calculate the confidence interval.

This interval helps us understand if the reduction in pain after hypnosis is statistically significant.

statistical hypothesis testing

Statistical hypothesis testing allows us to make decisions or inferences about a population based on sample data. In this study:

• The null hypothesis (H0) would be that hypnotism has no effect on pain reduction, meaning the mean difference is zero.
• The alternative hypothesis (H1) is that hypnotism does reduce pain, meaning the mean difference is not zero.

Our goal is to use the confidence interval to see if it includes zero. If zero is not in the interval, we reject the null hypothesis. This would suggest that hypnotism does indeed have an effect on reducing pain.

paired sample t-test

A paired sample t-test is used in this study because we have two measurements from the same group of subjects – before and after hypnosis. This type of test is perfect for comparing means from the same subjects under different conditions. Steps involved:

• Calculate the differences between each pair of observations.
• Find the mean and standard deviation of those differences.
• Compute the standard error using the standard deviation and the number of subjects.
• Use the t-distribution to determine the critical t-value for our sample size.
• Construct the confidence interval around the mean difference.

Interpretation of the paired sample t-test in this context helps determine if the observed changes from hypnotism are likely due to the treatment rather than random chance.