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# The article “The Functional Outcomes of Total Knee ISBN: 9780073401331 38

## Solution for problem 5E Chapter 5.2

Statistics for Engineers and Scientists | 4th Edition

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Problem 5E

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-by-Step Solution:

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).

Step 3 of 3

##### ISBN: 9780073401331

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