Question

the table shows the results when two different techniques are used to treat large and small kidney stones. each entry in the table gives the number of successful treatments and the total size of the treatment group. complete parts a through e.
(round to one decimal place as needed.)
treatment 1 treatment 2
small stones 92.7% 87.2%
large stones 73.3% 66.7%
b. which treatment has the higher success rate for small stones? for large stones?
treatment 1 has the higher success rate for small stones.
treatment 1 has the higher success rate for large stones.
c. now compute the success rate for both treatments when the cases of small and large stones are combined.
treatment 1 has a success rate of 78.1%.(round to one decimal place as needed.)
treatment 2 has a success rate of 82.4%.(round to one decimal place as needed.)
d. which treatment has the higher overall success rate for both types of stones?
treatment 2 has the higher success rate for both types of stones.
e. briefly explain why the results might seem surprising or paradoxical.
a. treatment 1 is more successful in each category (small stones and large stones), and yet treatment 2 is more successful overall. while both treatments had similar numbers of patients for each type of stone, they had different numbers of patients overall.
b. treatment 1 is more successful in each category (small stones and large stones), and yet treatment 2 is more successful overall. both treatments did worse on large stones.
c. treatment 1 is more successful in each category (small stones and large stones), and yet treatment 2 is more successful overall. while both treatments had similar numbers of patients overall, they had different numbers of patients for each type of stone.
d. treatment 1 is more successful in each category (small stones and large stones), and yet treatment 2 is more successful overall. both treatments did worse on small stones.

Explanation:

Step1: Analyze the success - rate data

We know that Treatment 1 has a higher success rate for both small stones (92.7%) and large stones (73.3%) when considered separately. But when we combine the cases of small and large stones, Treatment 1 has a success rate of 78.1% and Treatment 2 has a success rate of 82.4%.

Step2: Identify the reason for the paradox

The paradox occurs because while both treatments had similar numbers of patients overall, they had different numbers of patients for each type of stone. Treatment 2 likely had a more favorable distribution of patients between small and large stones when calculating the overall success - rate.

Answer:

C. Treatment 1 is more successful in each category (small stones and large stones), and yet Treatment 2 is more successful overall. While both treatments had similar numbers of patients overall, they had different numbers of patients for each type of stone.