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?Before6.42.67.710.511.75.84.32.8After6.72.47.48.18.66.43.92.7

Answer 1

The 95% confidence interval for the difference would be given by (-1.776;0.376)

We are 95% confidence that the true mean difference is between
`-1.776 \leq \mu_d \leq 0.376`. Since the confidence interval contains the value 0, we don't have enough evidence to conclude that hypnotism appear to be effective in reducing pain.

Explanation:

Previous concepts

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".

Let put some notation

x=test value before , y = test value after

x: 6.4 2.6 7.7 10.5 11.7 5.8 4.3 2.8

y: 6.7 2.4 7.4 8.1 8.6 6.4 3.9 2.7

The differences defined as
`d_i = y_i -x_i` and we got:

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

We can calculate the mean and the deviation for the sample with the following formulas:

`\bar d=(\sum_(i=1)^n d_i)/(n)`

`s_d =(\sum_(i=1)^n (d_i -\bar d)^2)/(n-1)`

`\bar d=-0.7` represent the sample mean for the difference

`\mu_d` population mean (variable of interest)

`s_d`=1.32 represent the sample standard deviation

n=8 represent the sample size

Confidence interval

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

`\bar d \pm t_(\alpha/2) *(s_d)/(√(n))` (1)

In order to calculate the critical value t we need to find 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(0.025,8)".And we see that
`t_(\alpha/2)=2.306`

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

`-0.7-2.306(1.32)/(√(8))=-1.776`

`-0.7+2.306(1.32)/(√(8))=0.376`

So on this case the 95% confidence interval for the difference would be given by (-1.776;0.376)

We are 95% confidence that the true mean difference is between
`-1.776 \leq \mu_d \leq 0.376`. Since the confidence interval contains the value 0, we don't have enough evidence to conclude that hypnotism appear to be effective in reducing pain.