Question

The confidence interval and test for comparison of proportions are frequently used in the media. The following is the part of news report: In contrast, the Democratic race tightened. Clinton is ahead by two points, edging Bernie Sanders by 48-46 percent. Last month, before Sanders won eight of the nine most recent contests, she had a 13-point advantage (55-42 percent). It was conducted by the Fox News under the joint direction of Anderson Robbins Research by telephone with live interviewers April 11-13, 2016. The number of samples used in the poll was 450 registered voters. a)Using the result of the survey in April, i.e. Clinton: 48% and Sanders: 46%, find two 95% confidence intervals for proportions of Clinton and Sanders, respectively. b)Based on your findings, you will answer whether or not you can predict the winner of Democratic Presidential Nomination? Explain. (Please ignore that Hillary was the winner of nomination and consider only this information for your decision.)

Answer 1

a.
i.
TRADITIONAL METHOD
given that,
possible chances (x)=216
sample size(n)=450
success rate ( p )= x/n = 0.48
I.
sample proportion = 0.48
standard error = Sqrt ( (0.48*0.52) /450) )
= 0.024
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 * 0.024
= 0.046
III.
CI = [ p ± margin of error ]
confidence interval = [0.48 ± 0.046]
= [ 0.434 , 0.526]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
possible chances (x)=216
sample size(n)=450
success rate ( p )= x/n = 0.48
CI = confidence interval
confidence interval = [ 0.48 ± 1.96 * Sqrt ( (0.48*0.52) /450) ) ]
= [0.48 - 1.96 * Sqrt ( (0.48*0.52) /450) , 0.48 + 1.96 * Sqrt ( (0.48*0.52) /450) ]
= [0.434 , 0.526]
-----------------------------------------------------------------------------------------------
interpretations:
1. We are 95% sure that the interval [ 0.434 , 0.526] 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
ii.
TRADITIONAL METHOD
given that,
possible chances (x)=207
sample size(n)=450
success rate ( p )= x/n = 0.46
I.
sample proportion = 0.46
standard error = Sqrt ( (0.46*0.54) /450) )
= 0.023
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 * 0.023
= 0.046
III.
CI = [ p ± margin of error ]
confidence interval = [0.46 ± 0.046]
= [ 0.414 , 0.506]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
possible chances (x)=207
sample size(n)=450
success rate ( p )= x/n = 0.46
CI = confidence interval
confidence interval = [ 0.46 ± 1.96 * Sqrt ( (0.46*0.54) /450) ) ]
= [0.46 - 1.96 * Sqrt ( (0.46*0.54) /450) , 0.46 + 1.96 * Sqrt ( (0.46*0.54) /450) ]
= [0.414 , 0.506]
-----------------------------------------------------------------------------------------------
interpretations:
1. We are 95% sure that the interval [ 0.414 , 0.506] 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
b.
from part (i), 95% confidence intervals for proportions of Clinton [ 0.434 , 0.526]
from part(ii),95% confidence intervals for proportions of sanders [ 0.414 , 0.506]
there is chance of winning Democratic Presidential Nomination is clinton have 43.4 to 52.6%.