A Study Was Conducted to Determine If Patients Recovering from Knee

A study was conducted to determine if patients recovering from knee surgery should receive physical therapy two or three times per week. Suppose $p _ { 1 }$ represents the proportion of patients who showed improvement after one month of therapy three times a week and $p _ { 2 }$ represents the proportion of patients who showed improvement after one month of therapy twice a week. A $90 \%$ confidence interval for $p _ { 1 } - p _ { 2 }$ is $( 0.12,0.38 )$ . Give an interpretation of this confidence interval.

A) We are $90 \%$ confident, based on the data, that patients who receive therapy three times per week show between $12 \%$ and $38 \%$ more improvement after one month than patients who receive therapy twice a week.
B) We know that $90 \%$ of all knee-surgery patients will show between $12 \%$ and $38 \%$ more improvement with therapy three times per week instead of twice a week.
C) We are $90 \%$ confident, based on the data, that the proportion of patients who show improvement with therapy three times per week is between $12 \%$ and $38 \%$ higher than the proportion who show improvement with therapy twice a week.
D) We know that $90 \%$ of all knee-surgery patients will show between $12 \%$ and $38 \%$ more improvement with therapy twice a week instead of three times per week.
E) We are $90 \%$ confident, based on the data, that the proportion of patients who show improvement with therapy three times per week is between $12 \%$ and $38 \%$ lower than the proportion who show improvement with therapy twice a week.