ADHD a researcher studying ADHD among teenagers obtains a simple random sample of 100 teenagers aged 13 to 17 and asks them whether or not they have ever been prescribed medication for ADHD. To say that the distribution of sample proportion (P cap), the sample proportion of teenagers who respond no, is approximately normal, how many more teenagers aged 13 to 17 does the researcher need to sample if
(a) 90% of all teenagers aged 13 to 17 have never been prescribed medication for ADHD?
•(b) 95% of all teenagers aged 13 to 17 have never been prescribed medication for ADHD?
To determine the number of more teenagers aged 13 to 17 that the researcher needs to sample, we can use the formula for the margin of error in estimating a population proportion.
The formula for the margin of error in estimating a population proportion is:
ME = Z * √(P hat * (1 - P hat) / n)
where:
ME is the margin of error,
Z is the z-value corresponding to the desired level of confidence,
P hat is the sample proportion,
n is the sample size.
For both parts (a) and (b), we need to calculate the margin of error and rearrange the formula to solve for the sample size (n). Then we can determine how many more teenagers the researcher needs to sample.
(a) When 90% of all teenagers aged 13 to 17 have never been prescribed medication for ADHD:
In this case, we want to find the sample size needed to estimate the proportion with a certain level of confidence. Let's assume a 95% confidence level, so the z-value is 1.96.
ME = 1.96 * √((0.9 * 0.1) / n)
Since we don't have the initial sample size (n), we can estimate it by setting ME as a proportion of the population:
ME = 1.96 * √((0.9 * 0.1) / n) = 0.01 * N
Here, N represents the total number of teenagers aged 13 to 17 in the population. Rearranging the equation, we can solve for n:
1.96 * √((0.9 * 0.1) / n) = 0.01 * N
Square both sides of the equation:
3.8416 * (0.9 * 0.1) / n = 0.0001 * N^2
0.3 / n = 0.0001 * N^2 / 3.8416
n = (0.3 * 3.8416) / (0.0001 * N^2)
By substituting the value of N (total number of teenagers aged 13 to 17 in the population) into the equation, you can calculate the required sample size.
(b) When 95% of all teenagers aged 13 to 17 have never been prescribed medication for ADHD:
The process is the same as in part (a). However, in this case, we need a higher confidence level, so the z-value is 1.96.
ME = 1.96 * √((0.95 * 0.05) / n) = 0.01 * N
0.95 * 0.05 / n = 0.0001 * N^2 / 3.8416
n = (0.95 * 0.05 * 3.8416) / (0.0001 * N^2)
Again, by substituting the value of N (total number of teenagers aged 13 to 17 in the population) into the equation, you can calculate the required sample size.