Howdy! I'm Professor Curtis of Aspire Mountain Academy here with more statistics homework help. Today we're going to learn how to find a binomial distribution probability for a combined test sample. Here's our problem statement: The probability of a randomly selected adult in one country being infected with a certain virus is 0.005. In tests for the virus, blood samples from 13 people are combined. What is the probability that the combined sample tests positive for the virus? Is it unlikely for such a combined sample to test positive? Note that the combined sample tests positive if at least one person has the virus.
To solve this problem I'm going to use StatCrunch. Inside StatCrunch, I'm going to pull up the binomial calculator. So to do that, I go to Stat –> Calculator and then select Binomial. Here's my binomial distribution calculator. Notice there are default values that are put in here. We're going to change those values because we have different numbers for our particular problem.
n is the number of people in our sample, which is 13 because we're taking 13 samples and combining them into one big sample. So this is the number that we use for n. p is the probability of success. Here we're defining success as being infected with a virus. That probability is given to us in the problem — 0.005. So we put that number in for p. Now, what do we put here for our random variable? Note that the combined sample tests positive if at least one person has the virus. This is statement from our problem statement. So we want the probability that at least one of those 13 are going to be affected. This means we want to say that x is going to be greater than or equal to one, because this means at least one person has the virus of the thirteen that we're testing.
I press Compute!, and out comes my probability that the combined sample will test positive. I'm asked to round to three decimal places. So here that would be 0.063. Put that answer in my answer field. Well done!
And now the second part of the problem asks, “Is it unlikely for such a combined sample to test positive?” Well, if we look at our answer options here, we can note the differences between them to select the proper answer. The beginning of each of the answer options says, “It is unlikely for such a combined sample to test positive” or “It is not unlikely for such a combined sample to test positive.” So it's either likely or unlikely. How do we judge that?
Well, look at the last part of our answer options. We're assessing whether it's greater than 0.05 or less than or equal to 0.05. So we're using 5% as a threshold value to test whether it's likely or unlikely the combined sample is going to test positive. A threshold value of 5% is a commonly used standard in many applications.
So let's look back at our probability. 0.063 is 6.3%. This is greater than 5%. Therefore, we're not going to select answer options B or D because these answers are options ending with “less than or equal than 0.05.” This means the correct answer must be either option A or C. So is it likely that the sample will test positive or unlikely that the sample will test positive? Well, this sample testing positive has a probability of 6.3%. It's greater than 5%. Therefore, it's going to be likely that we'll test positive. If the probability were less than 5%, then it would be unlikely. But our probability here is greater than 5%. Therefore, it is likely (or not unlikely). So the answer that we're going to select is answer option C. Fantastic!
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