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 a 95% confidence interval for the mean of the "before-after" differences. Does hypnotism appear to be effective in reducing pain? 4.9 Before After 7.8 6.4 9.9 7.7 12.0 8.6 9.2 8.4 9.2. 6.1 5.9 3.2 3.9 2.4 2.8 Construct a 95% confidence interval for the mean of the "before-after" differences. |< < (Round to two decimal places as needed.) Does hypnotism appear to be effective in reducing pain? O A. Yes, because the confidence interval includes zero. O B. Yes, because the confidence interval does not include zero and is entirely greater than zero. O C. No, because the confidence interval does not include zero and is entirely greater than zero. OD. No, because the confidence interval includes zero.

Answer 1

Step 1:

$\mu_{D}$ ≤ 0

$\mu_{D}$ > 0

Step 2: Test statistics

employee	Before	After	Diff (Bef-after)	Dev (diff - mean)	Sq deviation
1	7.8	6.4	1.4	-0.75	0.56
2	4.9	2.8	2.1	-0.05	0.00
3	9.9	7.7	2.2	0.05	0.00
4	12.0	8.6	3.4	1.25	1.56
5	9.2	8.4	0.8	-1.35	1.82
6	9.2	6.1	3.1	0.95	0.90
7	5.9	3.2	2.7	0.55	0.30
8	3.9	2.4	1.5	-0.65	0.42
Total	63	46	17.200	0.0000	5.5800

mean difference = 17.2 /8  = 2.15

sd difference = $SD_{D}$

$SD_{D} = \sqrt{\frac{\sum (x - \Bar{x})^{2}}{n-1}} = \sqrt{\frac{5.58}{7}}$

$SD_{D}$ = 0.8928

$t = \frac{\bar{x}_{D}}{\sqrt{SD_{D}/\sqrt{n}}} = \frac{2.15}{0.8928/\sqrt{8}}$

t = 6.811

tc​ = 1.895, for α = 0.05 and df=7

As t stat falls in the rejection area, we reject the Null hypothesis.

Hence we have sufficient evidence to believe that hypontism appears to be effective in reducing pain.

Confidence interval :

t value = TINV( 0.05, 7) = 2.365

E = 2.365 * (0.8928 / sqrt(8)) = 0.746

CI = 2.15 +/- 0.746

CI = 1.404 , 2.896

Correct option : B Yes, because the CI does not include zero and is entirely greater than zero.