The article “The Functional Outcomes of Total Knee Arthroplasty” (R. Kane, K. Saleh, et al.. Journal of Bone and Joint Surgery, 2005:1719-1724) reports that out of 10,501 surgeries, 859 resulted in complications within six months of surgery.

a. Find a 95% confidence interval for the proportion of surgeries that result in complications within six months.

b. Find a 99% confidence interval for the proportion of surgeries that result in complications within six months.

c. A surgeon claims that the rate of complications is less than 8.5%. With what level of confidence can this claim be made?

Step 1of 3:

An article report that out of 10,501 surgeries, 859 resulted in complications within six months of surgery.

We have to find

95% CI for the proportion of surgeries that result in complication within six months.99 % CI for the proportion of surgeries that result in complication within six months.The level of confidence for the claim that the rate of complications is less than 8.5%.Step 2 of 3:

The 95% CI for the proportion of surgeries that result in complications within six months.Let X be the number of success in an independent bernoulli trials with success probability P, so that X~B(n,P).

Lets define = n+4.

And =

Then a level 100(1-confidence interval for P is

Let X denote the number patients who resulted in complications within 6 months. And n be the total number of surgeries.

X = 859

n = 10,501

10501+ 4

= 10505

And

= 0.0819

Since here the Z-score is =

We can find from the standard normal table,

So the 95% confidence interval for the proportion of patients who resulted in complications with in six months.

( 0.0819 )

( 0.0767 , 0.0871 )

Therefore the 95% confidence interval for the proportion of patients who resulted in complication within six months is (0.0767, 0.0871).

(b) The 99% CI for the proportion of patients who resulted in complications within six months.

Since the Z- score is =

We can find the value from the standard normal table, .

So the 99% CI for the proportion of patients who resulted in complications within six month.

( 0.0819 )

( 0.0751 , 0.0899 )

Therefore the 99% confidence interval for the proportion of patients who resulted in complication within six months is (0.0751, 0.0899).