PROBLEM 6. The technology underlying hip replacements has changed as these surge
ID: 3310193 • Letter: P
Question
PROBLEM 6. The technology underlying hip replacements has changed as these surgeries have become more popular (over 250,000 in the U.S. in 2008). Starting in 2003, highly durable ceramic hips were marketed. Unfortunately, for many patients the increased durability has been counterbalanced by an increased incidence of squeaking. A certain study reported that out of 164 randomly selected individuals who received ceramic hips between 2003 and 2005, 41 of the hips developed squeaking. Construct and interpret a 98% (two-sided) confidence interval for a proportion of new type of hips that develop squeaking.
ANSWER:
Explanation / Answer
TRADITIONAL METHOD
given that,
possibile chances (x)=41
sample size(n)=164
success rate ( p )= x/n = 0.25
I.
sample proportion = 0.25
standard error = Sqrt ( (0.25*0.75) /164) )
= 0.0338
II.
margin of error = Z a/2 * (stanadard error)
where,
Za/2 = Z-table value
level of significance, = 0.02
from standard normal table, two tailed z /2 =2.326
margin of error = 2.326 * 0.0338
= 0.0786
III.
CI = [ p ± margin of error ]
confidence interval = [0.25 ± 0.0786]
= [ 0.1714 , 0.3286]
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DIRECT METHOD
given that,
possibile chances (x)=41
sample size(n)=164
success rate ( p )= x/n = 0.25
CI = confidence interval
confidence interval = [ 0.25 ± 2.326 * Sqrt ( (0.25*0.75) /164) ) ]
= [0.25 - 2.326 * Sqrt ( (0.25*0.75) /164) , 0.25 + 2.326 * Sqrt ( (0.25*0.75) /164) ]
= [0.1714 , 0.3286]
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interpretations:
1. We are 98% sure that the interval [ 0.1714 , 0.3286] contains the true population proportion
2. If a large number of samples are collected, and a confidence interval is created
for each sample, 98% of these intervals will contains the true population proportion
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