sl Millions of houses on the verge of foreclosure threaten to send homeownership
ID: 3275632 • Letter: S
Question
sl Millions of houses on the verge of foreclosure threaten to send homeownership to its lowest level in 50 ew industry estimates. Fresh projections say the rate could plummet to about 62% as y'as o2. In 2012, a survey of a random sample of 150 heads of households, 92 responded that they years, according to n owned their homes. a) Report (by hand) and compare a 90%, 95% and 99% confidence interval the homeownership rate for 2012. How does the width change as the confidence level increases? 90% confidence interval 95% confidence interval 99% confidence interval 90%, 95% and 99% confidence interval comparison: Confidence level Minitab output Manual calculation Width Lower bound, Upper bound) (Lower bound, Upper bound) 90% 95% 99% As the confidence level increases, the width of the confidence intervalExplanation / Answer
PART A.
When CI = 90%
TRADITIONAL METHOD
given that,
possibile chances (x)=92
sample size(n)=150
success rate ( p )= x/n = 0.613
I.
sample proportion = 0.613
standard error = Sqrt ( (0.613*0.387) /150) )
= 0.04
II.B9
margin of error = Z a/2 * (stanadard error)
where,
Za/2 = Z-table value
level of significance, = 0.1
from standard normal table, two tailed z /2 =1.645
margin of error = 1.645 * 0.04
= 0.065
III.
CI = [ p ± margin of error ]
confidence interval = [0.613 ± 0.065]
= [ 0.548 , 0.679] and width of interval is 0.131
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DIRECT METHOD
given that,
possibile chances (x)=92
sample size(n)=150
success rate ( p )= x/n = 0.613
CI = confidence interval
confidence interval = [ 0.613 ± 1.645 * Sqrt ( (0.613*0.387) /150) ) ]
= [0.613 - 1.645 * Sqrt ( (0.613*0.387) /150) , 0.613 + 1.645 * Sqrt ( (0.613*0.387) /150) ]
= [0.548 , 0.679]
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interpretations:
1. We are 90% sure that the interval [ 0.548 , 0.679] 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
When CI = 95%
CI = confidence interval
confidence interval = [ 0.613 ± 1.96 * Sqrt ( (0.613*0.387) /150) ) ]
= [0.613 - 1.96 * Sqrt ( (0.613*0.387) /150) , 0.613 + 1.96 * Sqrt ( (0.613*0.387) /150) ]
= [0.535 , 0.691] and width of interval is 0.156
When CI=99%
confidence interval = [ 0.613 ± 2.576 * Sqrt ( (0.613*0.387) /150) ) ]
= [0.613 - 2.576 * Sqrt ( (0.613*0.387) /150) , 0.613 + 2.576 * Sqrt ( (0.613*0.387) /150) ]
= [0.511 , 0.716] and width of interval is 0.205
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