Question

The proportion of physical therapy patients that continue to perform their neck injury exercises at home is 12%. a physical therapist is interested in encouraging patients to continue their exercises while at home to improve recovery. if the physical therapist chooses to study this further, for what sample size n will the sampling distribution of sample proportions have a standard deviation of 0.025?

Answer 1

The required sample size for the standard deviation of sample proportions to be 0.025, with a success proportion of 0.12, is calculated using the formula √[p(1-p)/n]. Solving this equation yields a sample size of 169.

Step-by-step explanation:

To determine the required sample size n for a sampling distribution of sample proportions with a standard deviation of 0.025, we use the formula for the standard deviation of a sample proportion, which is √[p(1-p)/n], where p is the proportion of success, which is given as 0.12 for the physical therapy patients performing exercises at home. We are looking for the sample size n that results in a standard deviation of 0.025. Plugging in the values we get:

0.025 = √[0.12(1 - 0.12)/n]

Square both sides to remove the square root:

0.000625 = 0.12(0.88)/n

Multiply both sides by n and then divide by 0.000625 to solve for n:

n = 0.12(0.88) / 0.000625

n = 0.1056 / 0.000625

n = 168.96

Since we can't have a fraction of a sample, we round up to the nearest whole number. Therefore, the required sample size n is 169.