Question

In a study of treatments for very painful​ "cluster" headaches, 158 patients were treated with oxygen and 140 other patients were given a placebo consisting of ordinary air. Among the158 patients in the oxygen treatment​ group, 109 were free from headaches 15 minutes after treatment. Among the 140 patients given the​ placebo, 28 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.

Identify the test statistic. Z= round to the 2nd decimal point

P = round to the 3rd decimal

Test the claim by constructing an appropriate confidence interval.

The 90% confidence interval is ? <( p 1 minus p 2 )< ?

.

Answer 1

Solution:-

a)

State the hypotheses. The first step is to state the null hypothesis and an alternative hypothesis.

Null hypothesis: P_Oxygen< P_Placebo
Alternative hypothesis: P_Oxygen > P_Placebo

Note that these hypotheses constitute a one-tailed test.

Formulate an analysis plan. For this analysis, the significance level is 0.10. The test method is a two-proportion z-test.

Analyze sample data. Using sample data, we calculate the pooled sample proportion (p) and the standard error (SE). Using those measures, we compute the z-score test statistic (z).

p = (p₁ * n₁ + p₂ * n₂) / (n₁ + n₂)

p = 0.459732

SE = sqrt{ p * ( 1 - p ) * [ (1/n₁) + (1/n₂) ] }
SE = 0.05785
z = (p₁ - p₂) / SE

z = 8.47

where p₁ is the sample proportion in sample 1, where p₂ is the sample proportion in sample 2, n₁ is the size of sample 1, and n₂ is the size of sample 2.

Since we have a one-tailed test, the P-value is the probability that the z-score is less than 8.47.

Thus, the P-value = 0.000.

Interpret results. Since the P-value (almost 0) is less than the significance level (0.05), we have to reject the null hypothesis.

From the above test we have sufficient evidence in the favor of the claim that the oxygen treatment is effective.

b) 90% confidence interval is C.I = (0.3947, 0.585).

C.I = (0.68987 - 0.20) + 1.645*0.05785

C.I = 0.48987 + 0.09516

C.I = (0.3947, 0.585)