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 151 individuals who received ceramic hips between 2003 and 2005, 10 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.) 033 (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 greater than the lower bound. We are 95% confident that the true proportion of all such artificial hip recipients who experience squeaking is less than the lower bound. You may need to use the appropriate table in the Appendix of Tables to answer this question.

Answer 1

SOLUTION:

From given data,

The technology underlying hip replacements has changed as these operations have become more popular (over 250,000 in the USA 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 151 individuals who received ceramic hips between 2003 and 2005, 10 of the hips developed squeaking.

X = 10

n =151

$p$ = X/n = 10/151 = 0.066

$q$ = 1-$p$ = 1-0.066 = 0.934

a Calculate a lower confidence bound at the 95% confidence interval for the true proportion of such hips that develop squeaking.

95% confidence interval

Confidence interval is 95%

95% = 95/100 = 0.95

$\alpha$ = 1 - Confidence interval = 1-0.95 = 0.05

$\alpha$/2 = 0.05 / 2

= 0.025

Z_$\alpha$/2 = Z_0.025 = 1.96

95% C.I

$p$$\pm$ Z_$\alpha$/2 * sqrt($p$$q$ / n)

0.066 $\pm$ 1.96 * sqrt(0.066*0.934 / 151)

0.066 $\pm$ 0.0396016

(0.066 - 0.0396016 , 0.066 + 0.0396016)

(0.026 , 0.105)

Lower confidence bound at the 95% confidence interval is 0.026.  

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 greater than the lower bound.

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