Question

Testing Claims About Proportions. In Exercises 7–22, test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim

Question:Headache Treatment In a study of treatments for very painful “cluster” headaches, 150 patients were treated with oxygen and 148 other patients were given a placebo consisting of ordinary air. Among the 150 patients in the oxygen treatment group, 116 were free from head- aches 15 minutes after treatment. Among the 148 patients given the placebo, 29 were free from headaches 15 minutes after treatment (based on data from “High-Flow Oxygen for Treatment of Cluster Headache,” by Cohen, Burns, and Goads by, Journal of the American Medical Association, Vol. 302, No. 22). We want to use a 0.01 significance level to test the claim that the oxygen treatment is effective.

a. Test the claim using a hypothesis test.

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

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

Short answer

a. The null hypothesis is rejected; thus, there is sufficient evidence to claim that the oxygen treatment is effective.

b. The 98% confidence interval is equal to\(\left( {0.467,0.687} \right)\).

c. Yes, the oxygen treatment is effective in curing cluster headaches.

Step-by-step solution

Given information

A sample of 150 patients was treated with oxygen, and among them, 116 were free from headaches 15 minutes after treatment. Another sample of 148 patients was given a placebo, and among them, 29 were free from headaches 15 minutes after treatment. The significance level is \(\alpha = 0.01\) .

Describe the hypotheses

It is claimed that oxygen treatment is effective; that is, the proportion of patients free from headache after oxygen treatment is greater than the proportion of patients free from headache after receiving a placebo.

Since the given claim does not have an equality sign, the following hypotheses are set up:

Null Hypothesis: The proportion of patients who are free from headache 15 minutes after oxygen treatment is equal to theproportion of patients who are free from headache 15 minutes after receiving placebo.

\({H_0}:{p_1} = {p_2}\)

Alternative hypothesis: The proportion of patients who are free from headache 15 minutes after oxygen treatment is greater than theproportion of patients who are free from headache 15 minutes after placebo treatment.

\({H_1}:{p_1} > {p_2}\)

The test is right-tailed.

Important values

Let \({\hat p_1}\) denote the sampleproportion of patients free from headache 15 minutes after oxygen treatment.

\(\begin{array}{c}{{\hat p}_1} = \frac{{{x_1}}}{{{n_1}}}\\ = \frac{{116}}{{150}}\\ = 0.7733\end{array}\)

Let\({\hat p_2}\) denote the sampleproportionof patients free from headache 15 minutes after receiving placebo.

\(\begin{array}{c}{{\hat p}_2} = \frac{{{x_2}}}{{{n_2}}}\\ = \frac{{29}}{{148}}\\ = 0.1959\end{array}\)

The sample size of patients who weretreated with oxygen treatment\(\left( {{n_1}} \right)\)is equal to 150.

The sample size of patients who weretreated with a placebo\(\left( {{n_2}} \right)\)is equal to 148.

The value of the pooled sample proportion is computed as follows:

\(\begin{array}{c}\bar p = \frac{{\left( {{x_1} + {x_2}} \right)}}{{\left( {{n_1} + {n_2}} \right)}}\,\\ = \frac{{\left( {116 + 29} \right)}}{{\left( {150 + 148} \right)}}\\ = 0.4866\end{array}\)

\(\begin{array}{c}\bar q = 1 - \bar p\\ = 1 - 0.4866\\ = 0.5134\end{array}\)

Find the test statistic

The test statistic is computed as follows:

\(\begin{array}{c}z = \frac{{\left( {{{\hat p}_1} - {{\hat p}_2}} \right) - \left( {{p_1} - {p_2}} \right)}}{{\sqrt {\left( {\frac{{\bar p\bar q}}{{{n_1}}} + \frac{{\bar p\bar q}}{{{n_2}}}} \right)} }}\\ = \frac{{\left( {0.7733 - 0.1959} \right) - 0}}{{\sqrt {\left( {\frac{{0.4866 \times 0.5134}}{{150}} + \frac{{0.4866 \times 0.5134}}{{148}}} \right)} }}\\ = 9.971\end{array}\)

The value of the test statistic is 9.971.

Referring to the standard normal distribution table, the critical value of z corresponding to\(\alpha = 0.01\)for a right-tailed test is equal to 2.33.

Referring to the standard normal distribution table, the corresponding p-value is equal to 0.0001.

Since the p-value is less than 0.01, the null hypothesis is rejected.

Conclusion of the test

a.

There is sufficient evidence to support the claim that theproportion of patients who were free from headache 15 minutes after oxygen treatment is greater than the proportion of patients who were free from headache 15 minutes after placebo treatment.

Find the confidence interval

b.

The general formula for the confidence interval of difference of proportion is written below:

\({\rm{Confidence}}\,\,{\rm{Interval}} = \left( {\left( {{{\hat p}_1} - {{\hat p}_2}} \right) - E\,\,,\,\,\left( {{{\hat p}_1} - {{\hat p}_2}} \right) + E} \right)\,\,\,\,\,\,\,\,...\left( 1 \right)\)\(\)

Where, E is the margin of error and has the following formula:

\(E = {z_{\frac{\alpha }{2}}} \times \sqrt {\left( {\frac{{{{\hat p}_1} \times {{\hat q}_1}}}{{{n_1}}} + \frac{{{{\hat p}_2} \times {{\hat q}_2}}}{{{n_2}}}} \right)} \)

For computing the confidence interval, first find the critical value\({z_{\frac{\alpha }{2}}}\).

The confidence level is 98% if the level of significance used in the one-railed test is 0.01 (part a.).

Thus, the value of the level of significance for the confidence interval becomes\(\alpha = 0.02\).

Hence,

\(\begin{array}{c}\frac{\alpha }{2} = \frac{{0.02}}{2}\\ = 0.01\end{array}\)

The value of\({z_{\frac{\alpha }{2}}}\)form the standard normal table is equal to 2.33.

Now, the margin of error (E) is equal to:

\(\begin{array}{c}E = {z_{\frac{\alpha }{2}}} \times \sqrt {\left( {\frac{{{{\hat p}_1} \times {{\hat q}_1}}}{{{n_1}}} + \frac{{{{\hat p}_2} \times {{\hat q}_2}}}{{{n_2}}}} \right)} \\ = 2.33 \times \sqrt {\left( {\frac{{0.7733 \times 0.2267}}{{150}} + \frac{{0.1959 \times 0.8041}}{{148}}} \right)} \\ = 0.1101\end{array}\)

Substitute the value of E in equation (1) as follows:

\(\begin{array}{c}{\rm{Confidence}}\,\,{\rm{Interval}} = \left( {\left( {{{\hat p}_1} - {{\hat p}_2}} \right) - E\,\,,\,\,\left( {{{\hat p}_1} - {{\hat p}_2}} \right) + E} \right)\\ = \left( {\left( {0.7733 - 0.1959} \right) - 0.1101\,\,,\,\,\left( {0.7733 - 0.1959} \right) + 0.1101} \right)\\ = \left( {0.467,0.687} \right)\end{array}\)

Thus, the 98% confidence interval for the difference between two proportions is\(\left( {0.467,0.687} \right)\).

Since the confidence interval does not include the value of 0 and includes all positive values, it can be concluded that the proportion of patients who are free from headache 15 minutes after oxygen treatment is greater than theproportion of patients who are free from headache 15 minutes after receiving placebo.

Effectiveness of the treatment

c.

From the results, it can be concluded thatthe oxygen treatment is effective because the proportion of patients free from headache after oxygen treatment is greater than the proportion of patients free from headache after receiving a placebo.