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!!!
you can do it that way, but why not work it as a continous problem, then round up to the next highest integer to get a whole person?
To solve this problem, we can use the concept of the binomial distribution. The binomial distribution is used to model situations where there are two possible outcomes (success or failure) for each trial and the probability of success is constant.
In this case, the trial is whether a person is approved for the study (success) or not (failure). The probability of success is 0.80, and we are interested in finding the sample size needed to ensure a probability of at least 0.939 of having at least 10 successes (patients) in the final approved group.
To work backwards and find the required initial sample size, we can use the concept of the complement of the probability. The complement of the probability of having at least 10 successes is the probability of having less than 10 successes. We can find this probability using the binomial cumulative distribution function.
Let's break down the problem step by step:
Step 1: Find the probability of having less than 10 successes in the final approved group.
To calculate this probability, we can use the binomial cumulative distribution function. In this case, we want to find P(X < 10), where X is the number of successes.
P(X < 10) = ∑ P(X = k) for k = 0 to 9
We can use a calculator or statistical software to calculate this probability. If you don't have access to such resources, there are also online calculators available specifically for the binomial distribution.
Step 2: Find the complement of the probability.
The complement of P(X < 10) is 1 - P(X < 10).
Step 3: Set up the inequality.
We want to find the sample size such that the complement of the probability is at least 0.939.
1 - P(X < 10) ≥ 0.939
Step 4: Find the sample size.
Now, you need to determine the sample size using the inequality you set up in step 3. You can use a trial and error approach, starting from a small sample size, and incrementally increase it until you find the minimum sample size that satisfies the inequality.
To simplify the calculations, you can use the binomial probability tables or again, use calculators or statistical software to find the appropriate sample size.
I hope this explanation helps you understand how to approach and solve this problem. If you have any further questions, feel free to ask!