An investigational drug study is being conducted. The initial screening phase needs to include enough people so that the final (approved) group has at least 10 patients. The probability of a person being approved for the study is .80. what initial sample size is needed to make sure with a .939 probability that the study will have at least 10 patients?
i don't know how to go about working this problem. i know its a discrete random variable problem and i know that i somehow have to work backwards, but i have NO idea how to set it up or work it. any and all guidance would be appreciated. thank you!!!
i don't expect you to work it for me, but i do need some help.
would this be a binomial-like should i consider it
applicable for study
not applicable for study
Yes, the binomial distribution is a discrete random variable problem.
Basically, if you screen N people, each having a success rate of 0.8. You would like to calculate the minimum candidates to have a minimum of 10 success with a probability of 0.939.
It is a very similar process to flipping coins, except the probability of success is 0.8 instead of 0.5.
The probability can be obtained by tabulation, but the a formula exists to calculate the probability of success of n times out of N candidates, with a probability of success of p.
P(n,N,p)=NCn*pN*(1-p)(N-n)
Let say, since the proability of success is 0.8, and 10 people are required, a good starting point is to screen 10/0.8=13 candidates.
So N=13, p=0.8,
P(10,13,0.8) = 0.246
P(11,13,0.8) = 0.268
P(12,13,0.8) = 0.179
P(13,13,0.8) = 0.055
Probability of retaining 10 or more candidates
= 0.246+0.268+0.179+0.055
= 0.747 < 0.939 required.
So you could screen 14 candidates and calculate the probability. Keep increasing the number of candidates until you get the probability of 0.939 or higher.
Total =
Oh yes, here's a link for more explanation:
http://stattrek.com/Lesson2/Binomial.aspx
Also, in case you are not familiar with the notation,
nCr is the combination function defined as:
nCr
= n!/((n-r)!r!)
=n(n-1)(n-2)...(n-r+1)/r!
For example,
13C10
=13*12*11/(1*2*3)
=286
thank you so much! i think i got it!
what was the answer you finally got
To solve this problem, you can use the concept of the cumulative binomial probability distribution. The probability mass function (PMF) of the binomial distribution is used to calculate the probability of getting a specific number of successes (in this case, the number of patients approved for the study) out of a fixed number of trials.
In this scenario, you want to find the initial sample size needed to ensure that the study will have at least 10 patients with a probability of 0.939. Let's break down the steps to solve the problem:
1. Set up the binomial distribution:
- Define the number of trials: n (unknown)
- Define the probability of success (a person being approved for the study): p = 0.80
- Define the desired cumulative probability: P(X >= 10) = 0.939
2. Calculate the complementary probability:
Since the binomial distribution is discrete (measuring the number of successes), finding the exact probability for P(X >= 10) is challenging. Instead, you can calculate the complementary probability of P(X < 10), which is equal to 1 - P(X >= 10).
3. Use the cumulative binomial distribution formula:
The cumulative binomial probability can be calculated using the formula:
P(X < x) = Σ (nCx * p^x * (1-p)^(n-x)), where Σ is the sum from x=0 to x=(n-1).
4. Iterate through different values of n:
Start with a small value of n, calculate the cumulative probability, and gradually increase n until the complementary probability is less than or equal to 0.061 (1 - 0.939).
5. Stop when the complementary probability reaches the desired level:
Once you find a value of n where P(X < 10) ≤ 0.061, you can stop because you have found the initial sample size needed to ensure a probability of at least 0.939 for the study to have at least 10 patients.
Please note that the calculations can be tedious and time-consuming to perform manually. Alternatively, you can use statistical software, such as Excel or online calculators to find the required sample size more efficiently.
I hope this explanation helps you understand the problem and guide you towards finding the solution.