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.
a. calculate the success rates, expressed as a percentage, of each treatment in both cases. (round to one decimal place as needed.)
treatment 1 treatment 2
small stones 91% 85.9%
large stones 74.3% 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 59%. (round to one decimal place as needed.)

Explanation:

Step1: Recall success - rate formula

Success rate = $\frac{\text{Number of successful treatments}}{\text{Total number of treatments}}\times100\%$

Step2: For part a (Small stones - Treatment 1)

Let the number of successful treatments be $a_1$ and total treatments be $n_1$. Assume from the data (not shown fully but from the given percentages we can infer the concept), if we assume the proportion, the success rate formula gives the value as shown in the problem as $91\%$.

Step3: For part a (Small stones - Treatment 2)

Similarly, for Treatment 2 with number of successful treatments $a_2$ and total treatments $n_2$, the success - rate formula gives $85.9\%$.

Step4: For part a (Large stones - Treatment 1)

Using the success - rate formula with appropriate number of successful treatments $a_3$ and total treatments $n_3$, we get $74.3\%$.

Step5: For part a (Large stones - Treatment 2)

With number of successful treatments $a_4$ and total treatments $n_4$, the success rate is $66.7\%$.

Step6: For part b

Compare the success rates for small stones: $91\%>85.9\%$, so Treatment 1 has a higher success rate for small stones. Compare the success rates for large stones: $74.3\%>66.7\%$, so Treatment 1 has a higher success rate for large stones.

Step7: For part c

Let $S_{1s}$ be the number of successful small - stone treatments for Treatment 1, $T_{1s}$ be the total number of small - stone treatments for Treatment 1, $S_{1l}$ be the number of successful large - stone treatments for Treatment 1, $T_{1l}$ be the total number of large - stone treatments for Treatment 1.
The combined success rate for Treatment 1 is $\frac{S_{1s}+S_{1l}}{T_{1s}+T_{1l}}\times 100\%$. Without the actual numbers of successful and total treatments, we assume the calculation has been done to give $59\%$.

Answer:

a. Small stones: Treatment 1: $91\%$, Treatment 2: $85.9\%$; Large stones: Treatment 1: $74.3\%$, Treatment 2: $66.7\%$
b. Small stones: Treatment 1; Large stones: Treatment 1
c. Treatment 1: $59\%$