Question

question 6 (10 points)
researchers conducted a study to determine which of two different treatments, a or b, is more effective in the treatment of atherosclerosis. the results of their experiment are given in the table. (a) which treatment appears to be more effective? why?
treatment a treatment b
effective 420 435
not effective 130 140
the data in the table do not take into account the seriousness of the case. the data shown in the next table show the effectiveness of each treatment for both mild and advanced cases of atherosclerosis.
mild atherosclerosis advanced atherosclerosis
treatment a treatment b treatment a treatment b
effective 310 95 110 340
not effective 80 20 50 120
(b) determine the proportion of mild cases of atherosclerosis that were effectively dealt with using treatment a. determine the proportion of mild cases of atherosclerosis that were effectively dealt with using treatment b.
(c) repeat part (b) for advanced cases of atherosclerosis to create a conditional distribution of effectiveness by treatment for each category of the disease.

Explanation:

Step1: Calculate proportion for mild cases - Treatment A

Proportion = $\frac{\text{Effective (Treatment A)}}{\text{Effective (Treatment A)}+\text{Not effective (Treatment A)}}$
$=\frac{310}{310 + 80}=\frac{310}{390}\approx0.795$

Step2: Calculate proportion for mild cases - Treatment B

Proportion = $\frac{\text{Effective (Treatment B)}}{\text{Effective (Treatment B)}+\text{Not effective (Treatment B)}}$
$=\frac{95}{95+20}=\frac{95}{115}\approx0.826$

Step3: Calculate proportion for advanced cases - Treatment A

Proportion = $\frac{\text{Effective (Treatment A)}}{\text{Effective (Treatment A)}+\text{Not effective (Treatment A)}}$
$=\frac{110}{110 + 50}=\frac{110}{160}=0.6875$

Step4: Calculate proportion for advanced cases - Treatment B

Proportion = $\frac{\text{Effective (Treatment B)}}{\text{Effective (Treatment B)}+\text{Not effective (Treatment B)}}$
$=\frac{340}{340+120}=\frac{340}{460}\approx0.739$

Answer:

• For mild cases: Treatment A proportion $\approx0.795$, Treatment B proportion $\approx0.826$
• For advanced cases: Treatment A proportion $= 0.6875$, Treatment B proportion $\approx0.739$