Question

The technology underlying hip replacements has changed as these operations have become more popular (over 250,000 in the United States in 2008). Starting in 2003, highly durable ceramic hips were marketed. Unfortunately, for too many patients the increased durability has been counterbalanced by an increased incidence of squeaking. An article reported that in one study of 142 individuals who received ceramic hips between 2003 and 2005, 8 of the hips developed squeaking.

(a) Calculate a lower confidence bound at the 95% confidence level for the true proportion of such hips that develop squeaking. (Round your answer to three decimal places.)


(b) Interpret the 95% confidence level used in (a).

We are 95% confident that the true proportion of all such artificial hip recipients who experience squeaking is less than the lower bound.

We are 95% confident that the true proportion of all such artificial hip recipients who experience squeaking is greater than the lower bound.    

  

Answer 1

a)

sample proportion, = 0.056
sample size, n = 142
Standard error, SE = sqrt(pcap * (1 - pcap)/n)
SE = sqrt(0.056 * (1 - 0.056)/142) = 0.0193

Given CI level is 95%, hence α = 1 - 0.95 = 0.05
α/2 = 0.05/2 = 0.025, Zc = Z(α/2) = 1.96

Margin of Error, ME = zc * SE
ME = 1.96 * 0.0193
ME = 0.0378

CI = (pcap - z*SE, pcap + z*SE)
CI = (0.056 - 1.96 * 0.0193 , 0.056 + 1.96 * 0.0193)
CI = (0.018 , 0.094)

Lower bound = 0.018

b)
We are 95% confident that the true proportion of all such artificial hip recipients who experience squeaking is greater than the lower bound.