Respuesta :
Answer:
A
[tex]t = 10.1[/tex]
B
[tex]p-value = p(t > 10.1)= 0.000[/tex]
C
[tex]0.4714 < p_1 - p_2 <0.6988[/tex]
D
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 [tex]n_1 = 140[/tex]
The number of patient which the oxygen cured is k = 113
The second sample size is [tex]n_2 = 158[/tex]
The number of patient that placebo cured is l = 35
The first sample proportion is
[tex]\r p_1 = \frac{ 113}{140 }[/tex]
[tex]\r p_1 = 0.8071[/tex]
The second sample proportion is
[tex]\r p_2 = \frac{ 35}{ 158 }[/tex]
[tex]\r p_2 = 0.222[/tex]
The null hypothesis is [tex]H_o : p_1 = p_2[/tex]
The alternative hypothesis is [tex]H_a : p_1 > p_2[/tex]
Let assume the level of significance be[tex]\alpha = 0.05[/tex]
Generally the pooled proportion is mathematically evaluated as
[tex]p = \frac{p1 * n1 + p2 * n2}{n1 + n2}[/tex]
substituting values
[tex]p = \frac{0.8071 * 140 + 0.222 * 158}{140 + 158}[/tex]
[tex]p = 0.4969[/tex]
Generally the standard error is mathematically represented
[tex]SE = \sqrt{ p(1- p ) * [ \frac{1}{n_1} + \frac{1}{n_1}] }[/tex]
substituting values
[tex]SE = \sqrt{ 0.4969(1- 0.4969 ) * [ \frac{1}{140} + \frac{1}{158}] }[/tex]
[tex]SE = 0.0580[/tex]
Generally the test statistics is mathematically represented as
[tex]t = \frac{ \r p_1 - \r p_2}{ SE}[/tex]
[tex]t = \frac{ 0.8071 -0.222}{ 0.0580}[/tex]
[tex]t = 10.1[/tex]
The p-value is from the normal distribution table as
[tex]p-value = p(t > 10.1)= 0.000[/tex]
given that [tex]t< \alpha[/tex] the null hypothesis is rejected
From the normal distribution table the critical value of [tex]\frac{\alpha }{2}[/tex] is
[tex]Z_{\frac{\alpha }{2} } = 1.96[/tex]
Generally the margin of error is mathematically the represented as
[tex]E = Z_{\frac{\alpha }{2} } * SE[/tex]
[tex]E = 1.96 * 0.0580[/tex]
[tex]E = 0.1137[/tex]
The 95% confidence interval is mathematically represented as
[tex](\r p_1 - \r p_2) - E < p_1 - p_2 < (\r p_1 - \r p_2) + E[/tex]
substituting value
[tex](0.8071 - 0.222) - 0.1137 < p_1 - p_2 < (0.8071 - 0.222) + 0.1137[/tex]
[tex]0.4714 < p_1 - p_2 <0.6988[/tex]
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
