Mathematics, 05.08.2020 09:01 wavy419

In a study of treatments for very painful "cluster" headaches, 140 patients were treated with oxygen and 158 other patients were given a placebo consisting of ordinary air. Among the 140 patients in the oxygen treatment group, 113 were free from headaches 15 minutes after treatment. Among the 158 patients given the placebo, 35 were free from headaches 15 minutes after treatment. Use a significance level to test the claim that the oxygen treatment is effective. A) Find test statistic z B) Find the P-value C) Construct the appropriate confidence interval D) determine if the oxygen treatment is effective

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Answer from: jlewis711777

t = 10.1

p-value = p(t 10.1)= 0.000

0.4714 < p_1 - p_2

The oxygen treatment is effective because

1 the is no 0 in the interval telling us that the treatments are different

2 the upper and the lower limit are positive tell us that the proportion of those treated by the oxygen treatment is greater

Step-by-step explanation:

From the question we are told that

The first sample size is n_1 = 140

The number of patient which the oxygen cured is k = 113

The second sample size is n_2 = 158

The number of patient that placebo cured is l = 35

The first sample proportion is

$\r p_1 = \frac{ 113}{140 }$

$\r p_1 = 0.8071$

The second sample proportion is

$\r p_2 = \frac{ 35}{ 158 }$

$\r p_2 = 0.222$

The null hypothesis is H_o : p_1 = p_2

The alternative hypothesis is H_a : p_1 p_2

Let assume the level of significance be $\alpha = 0.05$

Generally the pooled proportion is mathematically evaluated as

$p = \frac{p1 * n1 + p2 * n2}{n1 + n2}$

substituting values

$p = \frac{0.8071 * 140 + 0.222 * 158}{140 + 158}$

p = 0.4969

Generally the standard error is mathematically represented

$SE = \sqrt{ p(1- p ) * [ \frac{1}{n_1} + \frac{1}{n_1}] }$

substituting values

$SE = \sqrt{ 0.4969(1- 0.4969 ) * [ \frac{1}{140} + \frac{1}{158}] }$

SE = 0.0580

Generally the test statistics is mathematically represented as

$t = \frac{ \r p_1 - \r p_2}{ SE}$

$t = \frac{ 0.8071 -0.222}{ 0.0580}$

t = 10.1

The p-value is from the normal distribution table as

p-value = p(t 10.1)= 0.000

given that $t< \alpha$ the null hypothesis is rejected

From the normal distribution table the critical value of $\frac{\alpha }{2}$ is

$Z_{\frac{\alpha }{2} } = 1.96$

Generally the margin of error is mathematically the represented as

$E = Z_{\frac{\alpha }{2} } * SE$

E = 1.96 * 0.0580

E = 0.1137

The 95% confidence interval is mathematically represented as

$(\r p_1 - \r p_2) - E < p_1 - p_2 < (\r p_1 - \r p_2) + E$

substituting value

(0.8071 - 0.222) - 0.1137 < p_1 - p_2 < (0.8071 - 0.222) + 0.1137

0.4714 < p_1 - p_2

The oxygen treatment is effective because

1 the is no 0 in the interval telling us that the treatments are different

2 the upper and the lower limit are positive tell us that the proportion of those treated by the oxygen treatment is greater

Answer from: Quest

answer: 12

step-by-step explanation:

Answer from: Quest

9987

step-by-step explanation:

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In a study of treatments for very painful "cluster" headaches, 140 patients were treated with oxygen...