Question

In a study designed to test the effectiveness of magnets for treating back​ pain, 40 patients were given a treatment with magnets and also a sham treatment without magnets. Pain was measured using a scale from 0​ (no pain) to 100​ (extreme pain). After given the magnet​ treatments, the 40 patients had pain scores with a mean of 10.0 and a standard deviation of 2.8. After being given the sham​ treatments, the 40 patients had pain scores with a mean of 9.8 and a standard deviation of 2.32. Complete parts​ (a) through​ (c) below.

a. Construct the 99​% confidence interval estimate of the mean pain score for patients given the magnet treatment.

What is the confidence interval estimate of the population mean μ​?

__<μ​<__

b. Construct the 99% confidence interval estimate of the mean pain score for patients given the sham treatment.

What is the confidence interval estimate of the population mean μ​?

__<μ​<__

Answer 1

Solution :

Given that,

a) Point estimate = sample mean = = 10.0

sample standard deviation = s = 2.8

sample size = n = 40

Degrees of freedom = df = n - 1 = 40 -1 =39

At 99% confidence level the t is,

= 1 - 99%

= 1 - 0.99 = 0.01

/2 = 0.005

t _/2,df = t _{0.005, 39 =} 2.708

Margin of error = E = t_/2,df * (s /n)

= 2.708 * ( 2.8 / 40)

Margin of error = E = 1.2

The 99% confidence interval estimate of the population mean is,

- E < < + E

10.0 - 1.2 < < 10.0 + 1.2

(8.8 < < 11.2 )

b) Given that,

Point estimate = sample mean = = 9.8

sample standard deviation = s = 2.32

sample size = n = 40

Degrees of freedom = df = n - 1 = 40 - 1 = 39

At 99% confidence level the t is,

t _/2,df = 2.708

Margin of error = E = t_/2,df * (s /n)

= 2.708 * ( 2.32 / 40 )

Margin of error = E = 1.0

The 99% confidence interval estimate of the population mean is,

- E < < + E

9.8 - 1.0 < < 9.8 + 1.0

( 8.8 < < 10.8 )