Question

A random sample of size n=500 yielded p̂ =0.08
a) Construct a 95% confidence interval for p.
b) Interpret the 95% confidence interval.
c) Explain what is meant by the phrase "95% confidence interval."

Answer 1

TRADITIONAL METHOD
given that,
possibile chances (x)=40
sample size(n)=500
success rate ( p )= x/n = 0.08
I.
sample proportion = 0.08
standard error = Sqrt ( (0.08*0.92) /500) )
= 0.0121
II.
margin of error = Z a/2 * (stanadard error)
where,
Za/2 = Z-table value
level of significance, α = 0.05
from standard normal table, two tailed z α/2 =1.96
margin of error = 1.96 * 0.0121
= 0.0238
III.
CI = [ p ± margin of error ]
confidence interval = [0.08 ± 0.0238]
= [ 0.0562 , 0.1038]
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DIRECT METHOD
given that,
possibile chances (x)=40
sample size(n)=500
success rate ( p )= x/n = 0.08
CI = confidence interval
confidence interval = [ 0.08 ± 1.96 * Sqrt ( (0.08*0.92) /500) ) ]
= [0.08 - 1.96 * Sqrt ( (0.08*0.92) /500) , 0.08 + 1.96 * Sqrt ( (0.08*0.92) /500) ]
= [0.0562 , 0.1038]
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Answers:
a.
confidence interval = [0.0562 , 0.1038]

b.
interpretations:
1. We are 95% sure that the interval [ 0.0562 , 0.1038] contains the true population proportion
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 true population proportion

c.
phrase 95% of confidence interval means at p = 0.08 is (0.0562 , 0.1038)
If repeated samples were taken and the 95% confidence interval was computed for each sample,
95% of the intervals would contain the population mean.
A 95% confidence interval has a 0.95 probability of containing the population mean.
95% of the population distribution is contained in the confidence interval.