Question

 

A nursing student is studying how long it takes patients to receive pain relief from minor to moderate headache from two different pain relievers. Subjects are randomly and independently assigned to two groups where one is giving the leading product and the other given a new drug about to be released. Subjects then record the time it takes to realize relief from pain. Is there evidence to suggest the new drug works faster than the old?

Current 6.3 12.1 8.4 15.2 11.0 13.6 9.4 12.6 12.0 11.5 13.2 11.7 15.0 14.5

New 6.3 11.4 15.5 14.0 10.6 7.5 8.2 3.1 6.8 12.0 5.5 11.0 9.6 4.4

Answer 1

Solution

Let X = Time taken to receive relief from headache with the current product

      Y = Time taken to receive relief from headache with the new drug

Then, X ~ N(µ₁, σ₁²) and Y ~ N(µ₂, σ₂²), where σ₁² = σ₂² = σ², say and σ² is unknown.

Claim:

New drug works faster than the current drug; i.e., on an average, time to receive relief is shorter for the new drug than that for the current drug.   

Hypotheses:

Null: H₀: µ1 = µ2 Vs Alternative: H_A: µ1 > µ2 [claim]

Test Statistic:

t = (Xbar - Ybar)/{s√(2/n)} where

s² = (s₁² + s₂²)/2;

Xbar and Ybar are sample averages and s₁,s₂ are sample standard deviations based on n observations each on X and Y.

Calculations

Data

i	xi	yi
1	6.3	6.3
2	12.1	11.4
3	8.4	15.5
4	15	14.0
5	11	10.6
6	14	7.5
7	9	8.2
8	13	3.1
9	12	6.8
10	12	12.0
11	13	5.5
12	11.7	11.0
13	15	9.6
14	14.5	4.4

Summary of Excel calculations

n	14
Xbar	11.89
Ybar	8.99
s1	2.530571
s2	3.639202
s^2	9.823791
s	3.134293
tcal	2.447978
α	0.05
tcrit	1.705618
p-value	0.010714

α is assumed to be 0.05

Distribution, Significance Level, α , Critical Value and p-value:

Under H₀, t ~ t_{2n - 2}. Hence, for level of significance α%, Critical Value = upper α% point of t_{2n - 2} and p-value = P(t_{2n - 2} > tcal).

Assuming α = 0.05,

Using Excel Function: Statistical TINV TDIST the above are found to be as given in the above table.

Decision:

Since tcal > tcrit, or, equivalently, since p-value < α, H₀ is rejected.

Conclusion:

There is sufficient evidence to suggest that the claim is valid, implying that

New drug works faster than the current drug. Answer