Question

A physician compared two pain medications - medication 1 and medication 2, in terms of how many hours of relief each medication provides.
Medication 1 was given to 67 patients, and medication 2 was given to 80 patients.
The results are as follows:
Medication 1 - sample mean = 15.4 , sample standard deviation = 3
Medication 2 - sample mean = 13 , sample standard deviation = 7

(a) Obtain the 98% confidence interval for the difference between the mean relief times.

The 98% confidence interval is: (  ,  )

(b) At significance level 0.02, can we conclude that the two medications really differ in terms of the average relief time?

The test statistics =
Critical value =
Conclusion by critical value:
P-value =
The conclusion is (definite / borderline)

Answer 1

Answer: A physician compared two pain medications - medication 1 and medication 2, in terms of how many hours of relief each medication provides.

Solution:

n1 = 67, n2 = 80

x̄1 = 15.4, x̄2 = 13

s1 = 3, s2 = 7

(a) Obtain the 98% confidence interval for the difference between the mean relief times.

At 98% confidence interval, α = 0.02

df = [s1^2/n1 + s2^2/n2]^2 / [s1^2/n1]^2/n1-1 + [s2^2/n2]^2/n2-1

df = [3^2/67 + 7^2/80]^2 / [3^2/67]^2/67-1 + [7^2/80]^2/80-1

df = 111.0579

df = 111

t critical = t(α/2,df) = t(0.01,111)

t critical = 2.3604

Standard error, S.E:

SE = √s1^2/n1 + s^2/n2

SE = √3^2/67 + 7^2/80

SE = 0.8642

the 98% confidence interval:

CI = x̄1 - x̄2 ± t critical * SE

CI = 15.4 - 13 ± 2.3604 * 0.8642

CI = 2.4 ± 2.0398

CI = (0.3602, 4.4398)

Therefore, the 98% confidence interval for the difference between the mean relief times is between 0.3602 and 4.4398.

(b) At significance level 0.02, can we conclude that the two medications really differ in terms of the average relief time?

The hypothesis test:

Ho: μ1 = μ2

Ha: μ1 ≠ μ2

Test statistic t:

t = (x̄1 - x̄2)/SE

t = (15.4 - 13) / 0.8642

t = 2.7771

Test statistic t = 2.7771

Critical value of t:

t critical = 2.3604

Since test statistic t (2.7771) > t critical (2.3604)

We reject the null hypothesis Ho.

Conclusion:

Reject Ho. There is sufficient evidence to conclude that the two medications really differ in terms of the average relief time.

P-value:

df = 111

P-value = 0.0064

Since P-value (0.0064) < α (0.02) significance level.

We reject the null hypothesis Ho.

Conclusion:

Reject Ho. There is sufficient evidence to conclude that the two medications really differ in terms of the average relief time.