The residential energy consumption survey found in 2001 that 47% of American hou
ID: 3361062 • Letter: T
Question
The residential energy consumption survey found in 2001 that 47% of American households had internet access. A market survey organization repeat it this study and a certain town with 25,000 households, using a simple random sample of 500 households: 239 of the sample households had internet access. The percentage of households in the town with internet access is estimated as____; this estimate is likely to be off by___ or so
If possible find a 95% confidence interval for the percentage of all 25,000 households with internet access give the amount of the confidence interval or write not possible___.
Explanation / Answer
a.
sample one, x1 =500, n1 =25000, p1= x1/n1=0.02
sample two, x2 =239, n2 =25000, p2= x2/n2=0.01
p1 = 2%,p2 = 1%
b.
TRADITIONAL METHOD
given that,
sample one, x1 =500, n1 =25000, p1= x1/n1=0.02
sample two, x2 =239, n2 =25000, p2= x2/n2=0.01
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( (0.02*0.98/25000) +(0.01 * 0.99/25000))
=0.001
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 * 0.001
=0.002
III.
CI = (p1-p2) ± margin of error
confidence interval = [ (0.02-0.01) ±0.002]
= [ 0.008 , 0.013] = (0.8%,1.3%)
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DIRECT METHOD
given that,
sample one, x1 =500, n1 =25000, p1= x1/n1=0.02
sample two, x2 =239, n2 =25000, p2= x2/n2=0.01
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 = [ (0.02-0.01) ± 1.96 * 0.001]
= [ 0.008 , 0.013 ] = (0.8%,1.3%)
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interpretations:
1) we are 95% sure that the interval [ 0.008 , 0.013] 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