Question

.Ceramic hip replacements are very durable but can also be squeaky. The May 11, 2008 issue of the New York Times reported that in one study of randomly sampled 143 individuals who received ceramic hips beween 2003 and 2005, 10 of the hips developed squeaking. Calculate and interpret the 95% score CI for the true proportion of such hips that develop squeaking. (Include all 3 steps.)

Answer 1

Solution:

Given,

n = 143 ....... Sample size

x = 10 .......no. of successes in the sample

Point estimate of the true proportion is

= x/n = 10/143 = 0.07

Our aim is to construct 95% confidence interval.

c = 0.95

= 1- c = 1- 0.95 = 0.05

  /2 = 0.05 2 = 0.025 and 1- /2 = 0.975

Search the probability 0.975 in the Z table and see corresponding z value

= 1.96

Now , the margin of error is given by

E = /2 *  

= 1.96* [(0.07*(1 - 0.07)/143]

= 0.042

Now the confidence interval is given by

( - E)   ( + E)

(0.07 - 0. )   (0.07 + 0. )

0.028    0.112

Required 95% confidence Interval is (0.028 , 0.112 )

The meaning of 95% confidence interval : 95% confidence interval means that there is 95% probability that the true parameter will lie in the interval we constructed using 95% confidence level.

If we draw several samples of size n and we constructed the 95% confidence intervals using these samples , then 95% of these intervals will contain the true population parameter.

Here ,

There is 0.95 probability that the true proportion of such hips that develop squeaking will lie in the interval  (0.028 , 0.112 )