91Ó°ÊÓ

Use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal. 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

Subtract the 'After' values from the 'Before' values for each subject to get the differences. It is given as: A: 6.6 - 6.8 = -0.2 B: 6.5 - 2.4 = 4.1 C: 9.0 - 7.4 = 1.6 D: 10.3 - 8.5 = 1.8 E: 11.3 - 8.1 = 3.2 F: 8.1 - 6.1 = 2.0 G: 6.3 - 3.4 = 2.9 H: 11.6 - 2.0 = 9.6

02

- Calculate the mean of the differences

Add all the differences calculated previously and divide by the number of subjects. the Mean = \(\frac{-0.2 + 4.1 + 1.6 + 1.8 + 3.2 + 2.0 + 2.9 + 9.6}{8} = 3.75\)

03

- Calculate the standard deviation of the differences

Use the formula for standard deviation for the sample data: the standard deviation = \(\frac{\text{Sum of squared deviations}}{\text{number of subjects - 1}}\).First, calculate each squared deviation from the mean: = \[(-0.2 - 3.75)^2, (4.1 - 3.75)^2, (1.6 - 3.75)^2, (1.8 - 3.75)^2, (3.2 - 3.75)^2, (2.0 - 3.75)^2, (2.9 - 3.75)^2, (9.6 - 3.75)^2\] = \[15.0025, 0.1225, 4.6225, 3.8025, 0.3025, 2.7725, 0.7225, 34.2225\] Sum these values to get 61.57.Now divide it by (8-1):\(\frac{61.57}{7} ≈ 8.79571\).Finally, take the square root: \(\text{standard deviation} ≈ 2.9671\)

04

- Determine the critical value

For a 95% confidence level and degrees of freedom (df = 8 - 1 = 7), use the t-distribution table to find the critical value.The critical t-value for df = 7 and 95% confidence interval is roughly 2.365.

05

- Calculate the margin of error

Use the formula: \(E = t_{\frac{\text{critical}}{2}, df} * \frac{s}{\text{√n}}\)where: t is the critical value = 2.365s = standard deviation = 2.9671n = number of subjects = 8\(E = 2.365 * \frac{2.9671}{√8} ≈ 2.486\)

06

- Determine the confidence interval

Calculate the lower and upper limits of the confidence interval:Lower limit: Mean - Margin of Error = 3.75 - 2.486 = 1.264Upper limit: Mean + Margin of Error = 3.75 + 2.486 = 6.236So, the confidence interval is (1.264, 6.236).

07

- Interpretation

Check if the confidence interval includes zero. If zero is not included, the differences are statistically significant. Since the interval (1.264, 6.236) doesn't include zero, it suggests that hypnotism is effective in reducing pain.

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Key Concepts

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

paired sample data

When discussing the effectiveness of hypnotism in reducing pain, paired sample data is essential. Paired sample data occurs when we measure the same subjects under different conditions. In our study, we have two sets of pain measurements taken from the same subjects, before and after hypnosis.
By comparing these 'before' and 'after' pain values, we can precisely determine the impact of the hypnotic treatment on each subject. The paired sample approach eliminates individualistic differences because each person serves as their own control.
Here, we are looking at eight subjects and both their pain levels before and after hypnosis, represented in centimeters on a pain scale. Higher values imply more pain.

confidence interval

A confidence interval provides a range of values which is likely to contain a population parameter with a certain degree of confidence. In our context, it estimates the true mean difference in pain levels before and after hypnosis.
For example, we want a 95% confidence interval which suggests that we are 95% confident the true mean difference falls within the calculated range. This helps determine if hypnotism significantly reduces pain in a broader population, not just in our sample of eight.
Confidence intervals provide insight beyond a single estimate, offering a clearer picture of our data's reliability and variability.

mean difference

Calculating the mean difference is a crucial step. It represents the average change in pain levels due to hypnotism across all subjects. To find it, you subtract the 'after' pain values from the 'before' ones and then calculate the mean of these differences.
In our study, we get differences: -0.2, 4.1, 1.6, 1.8, 3.2, 2.0, 2.9, and 9.6. Adding these and dividing by the total number of subjects (8), we get a mean difference of 3.75. This positive mean difference indicates that, on average, pain levels decreased due to hypnotism.

standard deviation

Standard deviation measures the amount of variation or dispersion in our data. For paired sample data, it helps understand how much the differences in 'before' and 'after' pain scores deviate from the mean difference.
To calculate it, we first find the squared deviations from the mean difference for each subject, sum them up and divide by the number of subjects minus one (n-1), which gives us the variance. Taking the square root of this variance gives us the standard deviation.
In our example, the standard deviation of the differences is approximately 2.9671. This value gives us an idea of the spread of the differences around the mean difference.

t-distribution

The t-distribution comes into play when we have small sample sizes and need to make inferences about the population parameter. It is similar to the normal distribution but with thicker tails, accommodating more variability in smaller samples.
We use it to find the critical value needed to construct a confidence interval. The degrees of freedom (df) for our t-distribution is calculated as the sample size minus one (n-1). In our case, df = 8 - 1 = 7.
For a 95% confidence interval and 7 degrees of freedom, the critical t-value, found from the t-distribution table, is roughly 2.365. This value helps us determine the range in which the true mean difference likely falls.