n a large clinical trial, 397,876 children were randomly assigned to two groups.

Question

n a large clinical trial, 397,876 children were randomly assigned to two groups. The treatment group consisted of 198,978 children given a vaccine for a certain disease, and 38 of those children developed the disease. The other 198,898 children were given a placebo, and 126 of those children developed the disease. Consider the vaccine treatment group to be the first sample. Complete parts (a) through (d) below. a. Assume that a 0.01 significance level will be used to test the claim that p p2 Which is better: A hypothesis test or a confidence interval? b.In general, when dealing with inferences for two population proportions, which two of the following are equivalent: confidence interval method; P value method; critical value method? andare equivalent, in that they will always lead to the same conclusion. Both of these methods use a standard deviation based on (estimated values of the population proportions/the assumption that the two population proportions are equal), whereas the other method uses a standard deviation based on (estimated values of the population proportions/ the assumption that the two population proportions are equal) c. If a 0.01 significance level is to be used to test the claim that pi

Explanation / Answer

a.
Given that,
sample one, x1 =38, n1 =198978, p1= x1/n1=0.000191
sample two, x2 =126, n2 =198898, p2= x2/n2=0.000633
null, Ho: p1 = p2
alternate, H1: p1 < p2
level of significance, = 0.01
from standard normal table,left tailed z /2 =2.326
since our test is left-tailed
reject Ho, if zo < -2.326
we use test statistic (z) = (p1-p2)/(p^q^(1/n1+1/n2))
zo =(0-0.000633)/sqrt((0*0.999588(1/198978+1/198898))
zo =-6.875638
| zo | =6.875638
critical value
the value of |z | at los 0.01% is 2.326
we got |zo| =6.876 & | z | =2.326
make decision
hence value of | zo | > | z | and here we reject Ho
p-value: left tail - Ha : ( p < -6.8756 ) = 0
hence value of p0.01 > 0,here we reject Ho
ANSWERS
---------------
null, Ho: p1 = p2
alternate, H1: p1 < p2
test statistic: -6.875638
critical value: -2.326
decision: reject Ho
p-value: 0
b.
generally, we dealing with such problems two population proportion,hypothesis test and confidence interval method both using then we can get same conclusions
c.
TRADITIONAL METHOD
given that,
sample one, x1 =38, n1 =198978, p1= x1/n1=0.000191
sample two, x2 =126, n2 =198898, p2= x2/n2=0.000633
I.
standard error = sqrt( p1 * (1-p1)/n1 + p2 * (1-p2)/n2 )
where
p1, p2 = proportion of both sample observation
n1, n2 = sample size
standard error = sqrt( (0.000191*0.999809/198978) +(0.000633 * 0.999367/198898))
=0.000064
II.
margin of error = Z a/2 * (stanadard error)
where,
Za/2 = Z-table value
level of significance, = 0.01
from standard normal table,right tailed z /2 =2.33
margin of error = 2.33 * 0.000064
=0.00015
III.
CI = (p1-p2) ± margin of error
confidence interval = [ (0.000191-0.000633) ±0.00015]
= [ -0.000592 , -0.000293]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
sample one, x1 =38, n1 =198978, p1= x1/n1=0.000191
sample two, x2 =126, n2 =198898, p2= x2/n2=0.000633
CI = (p1-p2) ± sqrt( p1 * (1-p1)/n1 + p2 * (1-p2)/n2 )
where,
p1, p2 = proportion of both sample observation
n1,n2 = size of both group
a = 1 - (confidence Level/100)
Za/2 = Z-table value
CI = confidence interval
CI = [ (0.000191-0.000633) ± 2.33 * 0.000064]
= [ -0.000592 , -0.000293 ]
-----------------------------------------------------------------------------------------------
interpretations:
1) we are 99% sure that the interval [ -0.000592 , -0.000293] contains the difference between
true population proportion P1-P2
2) if a large number of samples are collected, and a confidence interval is created
for each sample, 99% of these intervals will contains the difference between
true population mean P1-P2

d.
first proportion is less thsn second proportion then there is sufficient evidence to support the rate of polio is less for children gives the vaccine than it is for children given placebo

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