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 minus−​after"differences. Does hypnotism appear to be effective in reducing​pain?

Answer 1

Answer:

$-0.402 < \mu_{before}- \mu_{after}$  

Since the confidence iterval contains the 0 we don't have enough evidence to conclude that hypnotism appear to be effective in reducing ​pain at 5 % of significance.

Step-by-step explanation:

Data given

Assuming the following dataset:

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

First we need to calculate first $d_i = x_{Before}-x_{after}$ and we got this:

d: -0.3, 0.2, 0.3, 2.4, 3.1, -0.6, 0.4, 0.1

The second step is calculate the mean difference  

$\bar d= \frac{\sum_{i=1}^n d_i}{n}=0.7$

The third step would be calculate the standard deviation for the differences, and we got:

$s_d =\frac{\sum_{i=1}^n (d_i -\bar d)^2}{n-1} =1.3202$

Confidence interval

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

The confidence interval for the mean is given by the following formula:  

$\bar d \pm t_{\alpha/2}\frac{s_d}{\sqrt{n}}$ (1)  

We need to calculate first the degrees of freedom given by:

$df=n-1=8-1=7$

Since the Confidence is 0.95 or 95%, the value of $\alpha=0.05$ and $\alpha/2 =0.025$, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,7)".And we see that $t_{\alpha/2}=\pm 2.36$  

Now we have everything in order to replace into formula (1):  

$0.7-2.36\frac{1.3202}{\sqrt{8}}=-0.402$  

$0.7+2.36\frac{1.3202}{\sqrt{8}}=1.802$  

So on this case the 95% confidence interval would be given by (-0.402;1.802)

$-0.402 < \mu_{before}- \mu_{after}$  

Since the confidence iterval contains the 0 we don't have enough evidence to conclude that hypnotism appear to be effective in reducing ​pain at 5 % of significance.