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Chapter 9: Problem 15

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? $$ \begin{array}{l|c|c|c|c|c|c|c|c|} \hline \text { Subject } & \text { A } & \text { B } & \text { C } & \text { D } & \text { E } & \text { F } & \text { G } & \text { H } \\ \hline \text { Before } & 6.6 & 6.5 & 9.0 & 10.3 & 11.3 & 8.1 & 6.3 & 11.6 \\ \hline \text { After } & 6.8 & 2.4 & 7.4 & 8.5 & 8.1 & 6.1 & 3.4 & 2.0 \\ \hline \end{array} $$

Short Answer

Expert verified
A 95% confidence interval can be constructed and used to determine if hypnotism reduces pain. If zero is not in the interval, hypnotism is effective.

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 鈥� 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.

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Most popular questions from this chapter

Large samples of women and men are obtained, and the hemoglobin level is measured in each subject. Here is the \(95 \%\) confidence interval for the difference between the two population means, where the measures from women correspond to population 1 and the measures from men correspond to population 2 : \(-1.76 \mathrm{~g} / \mathrm{dL}<\mu_{1}-\mu_{2}<-1.62 \mathrm{~g} / \mathrm{dL}\) a. What does the confidence interval suggest about equality of the mean hemoglobin level in women and the mean hemoglobin level in men? b. Write a brief statement that interprets that confidence interval. c. Express the confidence interval with measures from men being population 1 and measures from women being population \(2 .\)

In the largest clinical trial ever conducted, 401,974 children were randomly assigned to two groups. The treatment group consisted of 201,229 children given the Salk vaccine for polio, and 33 of those children developed polio. The other 200,745 children were given a placebo, and 115 of those children developed polio. If we want to use the methods of this section to test the claim that the rate of polio is less for children given the Salk vaccine, are the requirements for a hypothesis test satisfied? Explain.

In one segment of the TV series MythBusters, an experiment was conducted to test the common belief that people are more likely to yawn when they see others yawning. In one group, 34 subjects were exposed to yawning, and 10 of them yawned. In another group, 16 subjects were not exposed to yawning, and 4 of them yawned. We want to test the belief that people are more likely to yawn when they are exposed to yawning. a. Why can't we test the claim using the methods of this section? b. If we ignore the requirements and use the methods of this section, what is the \(P\) -value? How does it compare to the \(P\) -value of \(0.5128\) that would be obtained by using Fisher's exact test? c. Comment on the conclusion of the Mythbusters segment that yawning is contagious.

Researchers from the University of British Columbia conducted trials to investigate the effects of color on creativity. Subjects with a red background were asked to think of creative uses for a brick; other subjects with a blue background were given the same task. Responses were scored by a panel of judges and results from scores of creativity are given below. Higher scores correspond to more creativity. The researchers make the claim that "blue enhances performance on a creative task." a. Use a \(0.01\) significance level to test the claim that blue enhances performance on a creative task. b. Construct the confidence interval appropriate for the hypothesis test in part (a). What is it about the confidence interval that causes us to reach the same conclusion from part (a)? $$ \begin{array}{l|l} \hline \text { Red Background: } & n=35, \bar{x}=3.39, s=0.97 \\ \hline \text { Blue Background: } & n=36, \bar{x}=3.97, s=0.63 \\ \hline \end{array} $$

Two different simple random samples are drawn from two different populations. The first sample consists of 20 people with 10 having a common attribute. The second sample consists of 2000 people with 1404 of them having the same common attribute. Compare the results from a hypothesis test of \(p_{1}=p_{2}\) (with a \(0.05\) significance level) and a \(95 \%\) confidence interval estimate of \(p_{1}-p_{2}\).
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