Question

In a study of treatments for very painful​ "cluster" headaches, 160 patients were treated with oxygen and 146 other patients were given a placebo consisting of ordinary air. Among the 160 patients in the oxygen treatment​ group, 122 were free from headaches 15 minutes after treatment. Among the 146 patients given the​ placebo, 39 were free from headaches 15 minutes after treatment. Use a 0.05 significance level to test the claim that the oxygen treatment is effective. Complete parts​ (a) through​ (c) below.

Test the claim using a hypothesis test.

Consider the first sample to be the sample of patients treated with oxygen and the second sample to be the sample of patients given a placebo. What are the null and alternative hypotheses for the hypothesis​ test?

z=?

p-value=?

What is the conclusion based on the hypothesis​ test?

Test the claim by constructing an appropriate confidence interval. (90%)

What is the conclusion based on the confidence​ interval?

Based on the​ results, is the oxygen treatment​ effective?

Answer 1

To Test :-
H0 :- P1 = P2
H1 :- P1 > P2

p̂1 = 122 / 160 = 0.7625
p̂2 = 39 / 146 = 0.2671

Test Statistic :-
Z = ( p̂1 - p̂2 ) / √( p̂ * q̂ * (1/n1 + 1/n2) ))
p̂ is the pooled estimate of the proportion P
p̂ = ( x1 + x2) / ( n1 + n2)
p̂ = ( 122 + 39 ) / ( 160 + 146 )
p̂ = 0.5261
q̂ = 1 - p̂ = 0.4739
Z = ( 0.7625 - 0.2671) / √( 0.5261 * 0.4739 * (1/160 + 1/146) )
Z = 8.67

Test Criteria :-
Reject null hypothesis if Z > Z(α)
Z(α) = Z(0.05) = 1.6449
Z > Z(α) = 8.6683 > 1.6449, hence we reject the null hypothesis
Conclusion :- We Reject H0

Decision based on P value
P value = P ( Z > 8.6683 )
P value = 0
Reject null hypothesis if P value < α = 0.05
Since P value = 0 < 0.05, hence we reject the null hypothesis
Conclusion :- We Reject H0

(p̂1 - p̂2) ± Z(α/2) * √( ((p̂1 * q̂1)/ n1) + ((p̂2 * q̂2)/ n2) )
Z(α/2) = Z(0.1 /2) = 1.644854
Lower Limit = ( 0.7625 - 0.267123 )- Z(0.1/2) * √(((0.7625 * 0.2375 )/ 160 ) + ((0.267123 * 0.732877 )/ 146 ) = 0.4136
upper Limit = ( 0.7625 - 0.267123 )+ Z(0.1/2) * √(((0.7625 * 0.2375 )/ 160 ) + ((0.267123 * 0.732877 )/ 146 )) = 0.5772
90% Confidence interval is ( 0.4136 , 0.5772 )
( 0.4136 < ( P1 - P2 ) < 0.5772 )

Since the values of confidence interval is positive, we can conclude that to reject null hypothesis

Based on result, oxygen treatment is effective.