Question

A study was conducted to measure the effectiveness of hypnotism in reducing pain. The measurements are centimeters on a pain scale before and after hypnosis. Assume that the paired sample data are simple random samples and that the differences have a distribution that is approximately normal. Construct 95% confidence interval for the mean of the "before−after" differences. Does hypnotism appear to be effective in reducing pain?
Before 6.4 2.6 7.7 10.5 11.7 5.8 4.3 2.8
After 6.7 2.4 7.4 8.1 8.6 6.4 3.9 2.7

Answer 1

The confidence interval for mean of the "before-after" differences is $\fbox{(-0.4037,1.8037)}$

Further explanation:

Find the difference between the before pain and the after pain.

Difference = before-after

Kindly refer to the Table for the difference of between the before and after pain.

Sum of difference = $5.6$

Total number of observation = $8$

Mean of difference = $0.7$

Sample standard deviation $s$ = $1.3201$

Level of significance = $5\%$

Formula for confidence interval = $\left( \bar{X} \pm t_{n-1, \frac{\alpha}{2}\%} \frac{s}{\sqrt{n}} \right)$

confidence interval = $\left( 0.7 \pm t_{8-1, \frac{5}{2}\%} \frac{1.3201}{\sqrt{8}} \right)$

confidence interval = $\left( 0.7 \pm t_{7, \frac{5}{2}\%} \frac{1.3201}{\sqrt{8}} \right)$

From the t-table.

The value of $t_{7, \frac{5}{2}\%$=$2.365$

Confidence interval = $( 0.7 \pm 2.365}\times \frac{1.3201}{\sqrt{8}}) \right)$

Confidence interval = $\left( 0.7 - 2.365}\times \0.4667,0.7 + 2.365}\times \0.4667) \right$

Confidence interval = $(0.7-1.1037,0.7+1.1037)$

Confidence interval = $\fbox{(-0.4037,1.8037)}$

The $95\%$ confidence interval tells us about that $95\%$ chances of the true mean or population mean lies in the interval.

Yes, the hypnotism appear to be effective in reducing pain as confidence interval include includes the positive deviation from the mean.

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Answer Details:

Grade: College Statistics

Subject: Mathematics

Chapter: Confidence Interval

Keywords:

Probability, Statistics, Speed dating, Females rating, Confidence interval, t-test, Level of significance , Normal distribution, Central Limit Theorem, t-table, Population mean, Sample mean, Standard deviation, Symmetric, Variance.

Answer 2

Answer:

The 95% confidence interval for the mean of the “before-after” difference is (-0.4039,1.8039)

No, Hypnotism doesn’t appear to be effective in reducing pain.

Further explanation:

Given: The table of measure the effectiveness of hypnotism in reducing pain.

Before : 6.4    2.6    7.7   10.5    11.7    5.8    4.3    2.8

After    : 6.7    2.4    7.4     8.1     8.6    6.4    3.9    2.7

we make the table of difference between “before-after”

(Before-After) :

6.4-6.7  2.6-2.4  7.7-7.4  10.5-8.1  11.7-8.6  5.8-6.4  4.3-3.9  2.8-2.7

   -0.3        0.2       0.3         2.4         3.1        -0.6       0.4         0.1

Now, we find the sample mean and sample standard deviation of above table.

$\text{Sample Mean, }\bar{x}=\dfrac{\text{Sum of number}}{\text{number of observation}}$

$=\dfrac{-0.3+0.2+0.3+2.4+3.1-0.6+0.4+0.1}{8}$

$=\dfrac{5.6}{8}=0.7$

$\text{Sample Standard deviation, s} = \sqrt{\dfrac{(x-\bar{x})^2}{n-1}}$

$=\sqrt{\dfrac{(-0.3-0.7)^2+(0.2-0.7)^2+(0.3-0.7)^2 ...+(0.1-0.7)^2}{8-1}}$

$=\dfrac{12.2}{7}=1.3201$

For 95% confidence interval $\mu_d$ using t-distribution

$\text{Marginal Error, E}=t_{\frac{\alpha}{2},df}\times \dfrac{s}{\sqrt{n}}$

Where,

• $t_{\frac{\alpha}{2},df}$ is critical value.
• alpha is significance level, $\alpha=1-0.95=0.05$
• df is degree of freedom for t-distribution, df=n-1 =7
• s is sample standard deviation, s=1.3201
• n is sample size, n=8

For critical value,

$\Rightarrow t_{\frac{\alpha}{2},df}$

$\Rightarrow t_{\frac{0.05}{2},7}$

$\Rightarrow t_{0.025,7}$

using t-distribution two-tailed table,

$t_{0.025,7}=2.365$

Substitute the values into formula and calculate E

$E=2.65\times \dfrac{1.301}{\sqrt{8}}$

Therefore, Marginal error, E=1.1039

95% confidence interval given by:

$=\bar{x}\pm E$

$=0.7\pm 1.1039$

• For lowest value of interval: 0.7-1.1039 = -0.4039
• For largest value of interval: 0.7+1.1039 = 1.8039

Therefore, 95% confidence interval using t-distribution: (-0.4039,1.8039)

This interval contains 0

Therefore, Hypnotism doesn’t appear to be effective in reducing pain.  

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Keywords:  

T-distribution, Sample mean, sample standard deviation, Critical value of t, degree of freedom, t-test, confidence interval, significance level.