Any hypothesis test must be performed at the 5% level of significance. In the gr

Question

Any hypothesis test must be performed at the 5% level of significance.

In the great city of chicago a group of engineers decided to check the effectivness of a certine type of flu Vaccine. They gave the vaccine to a randomly selected 2144 people. After the flu season passed. only 300 people who had vaccinations had flu, compared to 480 of 2140 randomly slelected people who did not have vaccination. Determine a 95% confidence interval for the difference in proportions of flu. Do you think the vaccine was effective?

Explanation / Answer

TRADITIONAL METHOD
given that,
sample one, x1 =300, n1 =2144, p1= x1/n1=0.1399
sample two, x2 =480, n2 =2140, p2= x2/n2=0.2243
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( (0.1399*0.8601/2144) +(0.2243 * 0.7757/2140))
=0.0117
II.
margin of error = Z a/2 * (stanadard error)
where,
Za/2 = Z-table value
level of significance, = 0.05
from standard normal table, two tailed z /2 =1.96
margin of error = 1.96 * 0.0117
=0.023
III.
CI = (p1-p2) ± margin of error
confidence interval = [ (0.1399-0.2243) ±0.023]
= [ -0.1074 , -0.0614]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
sample one, x1 =300, n1 =2144, p1= x1/n1=0.1399
sample two, x2 =480, n2 =2140, p2= x2/n2=0.2243
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 = [ (0.1399-0.2243) ± 1.96 * 0.0117]
= [ -0.1074 , -0.0614 ]
-----------------------------------------------------------------------------------------------
interpretations:
1) we are 95% sure that the interval [ -0.1074 , -0.0614] 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, 95% of these intervals will contains the difference between
true population mean P1-P2

as we see zero is not lies in the interval we concllude that people who undergo vaccination is effective
compared to who are not

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