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Get Full Access to Applied Statistics And Probability For Engineers - 6 Edition - Chapter 2.5 - Problem 131e
Get Full Access to Applied Statistics And Probability For Engineers - 6 Edition - Chapter 2.5 - Problem 131e

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# An article in the British Medical Journal [“Comparison of

ISBN: 9781118539712 55

## Solution for problem 131E Chapter 2.5

Applied Statistics and Probability for Engineers | 6th Edition

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Problem 131E

Problem 131E

An article in the British Medical Journal [“Comparison of treatment of renal calculi by operative surgery, percutaneous nephrolithotomy, and extracorporeal shock wave lithotripsy” (1986, Vol. 82, pp. 879–892)] provided the following discussion of success rates in kidney stone removals. Open surgery had a success rate of 78% (273/350) and a newer method, percutaneous nephrolithotomy (PN), had a success rate of 83% (289/350). This newer method looked better, but the results changed when stone diameter was considered. For stones with diameters less than 2 centimeters, 93% (81/87) of cases of open surgery were successful compared with only 83% (234/270) of cases of PN. For stones greater than or equal to 2 centimeters, the success rates were 73% (192/263) and 69% (55/80) for open surgery and PN, respectively. Open surgery is better for both stone sizes, but less successful in total. In 1951, E. H. Simpson pointed out this apparent contradiction (known as Simpson’s paradox), and the hazard still persists today. Explain how open surgery can be better for both stone sizes but worse in total.

Step-by-Step Solution:

Solution :

Step 1 of 1:

From the given information the success rate in kidney stone removals table is given below.

The the total open surgery had success rate is

 Type Percent Fraction Open surgery 78% PN 83%

Then stone diameter is less than 2 centimeters.

The table is given below.

 Type Percent Fraction Open surgery 93% PN 83%

Our goal is:

We need to explain how open surgery can be better for both stone sizes but worse in total.

Now the stone diameter is greater or equals to 2 centimeters.

Then the table is given below.

 Type Percent Fraction Open surgery 73% PN 69%

Now we have to explain how open surgery can be better for both stone sizes but worse in total.

The open surgery table is given below.

 Success Failure Sample size Sample percentage Conditional success rate Large stone 192 71 263 75% 73% Small stone 81 6 87 25% 93% Total 273 77 350 100% 78%

The PN surgery table is given below.

 Success Failure Sample size Sample percentage Conditional success rate Large stone 55 25 80 23% 69% Small stone 234 36 270 77% 83% Total 289 61 350 100% 83%

From the above table there are a lot more open surgeries the stone have diameter is less than 2 centimeters, so when the stones have a diameter of less than 2 centimeters then there are a lot of PN’s.

Here the total success rate is depends on the success rate for each stone size, but same as the probability group.

So this the weighted average of the group success rate weighted by the group size is

P(overall success) = P.

So open surgery, the large stone has a smaller success rate while for PN, the small stone has a larger rate.

Step 2 of 1