Question

1. In a poll, 191 out of a sample of 357 likely Connecticut voters indicate they intend to vote to lega marijuana. Find a. A conservative 99% confidence interval for the proportion of all likely Connecticut voters who are in favor of this initiative. b. An approximate 95% confidence interval for the same proportion. C. An improved 90% confidence interval for the same proportion.

Answer 1

1.
a.
TRADITIONAL METHOD
given that,
possible chances (x)=191
sample size(n)=357
success rate ( p )= x/n = 0.535
I.
sample proportion = 0.535
standard error = Sqrt ( (0.535*0.465) /357) )
= 0.026
II.
margin of error = Z a/2 * (standard error)
where,
Za/2 = Z-table value
level of significance, α = 0.01
from standard normal table, two tailed z α/2 =2.576
margin of error = 2.576 * 0.026
= 0.068
III.
CI = [ p ± margin of error ]
confidence interval = [0.535 ± 0.068]
= [ 0.467 , 0.603]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
possible chances (x)=191
sample size(n)=357
success rate ( p )= x/n = 0.535
CI = confidence interval
confidence interval = [ 0.535 ± 2.576 * Sqrt ( (0.535*0.465) /357) ) ]
= [0.535 - 2.576 * Sqrt ( (0.535*0.465) /357) , 0.535 + 2.576 * Sqrt ( (0.535*0.465) /357) ]
= [0.467 , 0.603]
-----------------------------------------------------------------------------------------------
interpretations:
1. We are 99% sure that the interval [ 0.467 , 0.603] contains the true population proportion
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 true population proportion
b.
TRADITIONAL METHOD
given that,
possible chances (x)=191
sample size(n)=357
success rate ( p )= x/n = 0.535
I.
sample proportion = 0.535
standard error = Sqrt ( (0.535*0.465) /357) )
= 0.026
II.
margin of error = Z a/2 * (standard 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 * 0.026
= 0.052
III.
CI = [ p ± margin of error ]
confidence interval = [0.535 ± 0.052]
= [ 0.483 , 0.587]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
possible chances (x)=191
sample size(n)=357
success rate ( p )= x/n = 0.535
CI = confidence interval
confidence interval = [ 0.535 ± 1.96 * Sqrt ( (0.535*0.465) /357) ) ]
= [0.535 - 1.96 * Sqrt ( (0.535*0.465) /357) , 0.535 + 1.96 * Sqrt ( (0.535*0.465) /357) ]
= [0.483 , 0.587]
-----------------------------------------------------------------------------------------------
interpretations:
1. We are 95% sure that the interval [ 0.483 , 0.587] 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.
TRADITIONAL METHOD
given that,
possible chances (x)=191
sample size(n)=357
success rate ( p )= x/n = 0.535
I.
sample proportion = 0.535
standard error = Sqrt ( (0.535*0.465) /357) )
= 0.026
II.
margin of error = Z a/2 * (standard error)
where,
Za/2 = Z-table value
level of significance, α = 0.1
from standard normal table, two tailed z α/2 =1.645
margin of error = 1.645 * 0.026
= 0.043
III.
CI = [ p ± margin of error ]
confidence interval = [0.535 ± 0.043]
= [ 0.492 , 0.578]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
possible chances (x)=191
sample size(n)=357
success rate ( p )= x/n = 0.535
CI = confidence interval
confidence interval = [ 0.535 ± 1.645 * Sqrt ( (0.535*0.465) /357) ) ]
= [0.535 - 1.645 * Sqrt ( (0.535*0.465) /357) , 0.535 + 1.645 * Sqrt ( (0.535*0.465) /357) ]
= [0.492 , 0.578]
-----------------------------------------------------------------------------------------------
interpretations:
1. We are 90% sure that the interval [ 0.492 , 0.578] contains the true population proportion
2. If a large number of samples are collected, and a confidence interval is created
for each sample, 90% of these intervals will contains the true population proportion