Question

In a study of treatments for very painful​ "cluster" headaches, 154 patients were treated with oxygen and 159 other patients were given a placebo consisting of ordinary air. Among the 154 patients in the oxygen treatment​ group,112 were free from headaches 15 minutes after treatment. Among the 159 patients given the​ placebo, 32 were free from headaches 15 minutes after treatment. Use a 0.01 significance level to test the claim that the oxygen treatment is effective. Complete parts​ (a) through​ (c) below.

a. 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?

A.

Upper H 0H0​:

p 1p1not equals≠p 2p2

Upper H 1H1​:

p 1p1equals=p 2p2

B.

Upper H 0H0​:

p 1p1greater than or equals≥p 2p2

Upper H 1H1​:

p 1p1not equals≠p 2p2

C.

Upper H 0H0​:

p 1p1equals=p 2p2

Upper H 1H1​:

p 1p1greater than>p 2p2

D.

Upper H 0H0​:

p 1p1equals=p 2p2

Upper H 1H1​:

p 1p1not equals≠p 2p2

E.

Upper H 0H0​:

p 1p1less than or equals≤p 2p2

Upper H 1H1​:

p 1p1not equals≠p 2p2

F.

Upper H 0H0​:

p 1p1equals=p 2p2

Upper H 1H1​:

p 1p1less than<p 2p2

Identify the test statistic.

z=

​(Round to two decimal places as​ needed.)

Identify the​ P-value.

​P-value=

​(Round to three decimal places as​ needed.)

What is the conclusion based on the hypothesis​ test?

The​ P-value is

▼

greater than

less than

the significance level of

alphaαequals=0.010.01​,

so

▼

fail to reject

reject

the null hypothesis. There

▼

is sufficient

is not sufficient

evidence to support the claim that the cure rate with oxygen treatment is higher than the cure rate for those given a placebo.

b. Test the claim by constructing an appropriate confidence interval.

The

9898​%

confidence interval is

nothingless than<left parenthesis p 1 minus p 2 right parenthesisp1−p2less than<nothing.

​(Round to three decimal places as​ needed.)

What is the conclusion based on the confidence​ interval?

Because the confidence interval limits

▼

include

do not include

​0, it appears that the two cure rates are

▼

equal.

not equal.

Because the confidence interval limits include

▼

only negative

only positive

positive and negative

​values, it appears that the cure rate is

▼

higher

lower

the same

for the oxygen treatment than for the placebo.

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

A.

The results suggest that the oxygen treatment is not effective in curing​ "cluster" headaches because the cure rate for the oxygen treatment appears to be lower than that of the placebo.

B.

The results suggest that the oxygen treatment is not effective in curing​ "cluster" headaches because the cure rates appear to be the same.

C.

The results suggest that the oxygen treatment is effective in curing​ "cluster" headaches.

D.

The results are inconclusive.

Answer 1

For sample 1, we have that the sample size is N1​=154, the number of favorable cases is X1​=112, so then the sample proportion is

For sample 2, we have that the sample size is N2​=159, the number of favorable cases is X2​=32, so then the sample proportion is

The value of the pooled proportion is computed as

Also, the given significance level is α=0.01.

Null and Alternative Hypotheses

The following null and alternative hypotheses need to be tested:

Ho: ​

Ha: ​

This corresponds to a right-tailed test, for which a z-test for two population proportions needs to be conducted.

Rejection Region

Based on the information provided, the significance level is α=0.01, and the critical value for a right-tailed test is zc​=2.33.

The rejection region for this right-tailed test is R={z:z>2.33}

Test Statistics

The z-statistic is computed as follows:

Decision about the null hypothesis

Since it is observed that z=9.335>zc​=2.33, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value is p=0, and since p=0<0.01, it is concluded that the null hypothesis is rejected.

Conclusion

It is concluded that the null hypothesis Ho is rejected. Therefore, there is enough evidence to claim that population proportion p1​ is greater than p2​, at the 0.01 significance level.

OPTION C -

The results suggest that the oxygen treatment is effective in curing​ "cluster" headaches.

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