Based on data collected by the National Center for Health Statistics and made available to the public in
the Sample Adult database, an estimate of the percentage of adults who indicated to have at some
point of their life been tested for HIV is 32 percent of U.S. adults in the database. Consider a simple
random sample of 15 adults selected at that time. Find the probability that the number of adults who
have been tested for HIV in the sample would be:
(b) Less than five (c) Between five and nine, inclusive
(d) More than five, but less than 10 (e) six or more.
To calculate the probabilities requested, we need to use the concept of the binomial distribution. The binomial distribution is used when we have a fixed number of independent trials with two possible outcomes, and we want to find the probability of a particular number of successes.
In this case, we have a random sample of 15 adults, and we want to find the probability of a specific number of adults who have been tested for HIV.
Firstly, let's determine the parameters for the binomial distribution:
n = 15 (number of trials)
p = 0.32 (probability of success, which is the percentage of adults who have been tested for HIV)
Now, let's calculate the probabilities for the given scenarios:
(b) Less than five:
We need to find the probability of having 0, 1, 2, 3, or 4 adults who have been tested for HIV. We can calculate this by summing up the individual probabilities for each scenario using the binomial probability formula:
P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
You can calculate each individual probability using the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
where "n choose k" represents the number of combinations of n items taken k at a time.
In this case, "n choose k" is calculated as:
(n choose k) = n! / (k! * (n - k)!)
Using this formula, you can calculate the probability of each scenario and sum them up to find P(X < 5).
(c) Between five and nine, inclusive:
We need to find the probability of having 5, 6, 7, 8, or 9 adults who have been tested for HIV. Similar to the previous scenario, we can calculate this by summing up the individual probabilities for each scenario.
P(5 ≤ X ≤ 9) = P(X = 5) + P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9)
(d) More than five but less than ten:
We need to find the probability of having more than 5 but less than 10 adults who have been tested for HIV. This involves calculating the probability of 6, 7, 8, or 9 adults who have been tested for HIV.
P(6 < X < 10) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9)
(e) Six or more:
We need to find the probability of having 6, 7, 8, 9, 10, 11, 12, 13, 14, or 15 adults who have been tested for HIV.
P(X ≥ 6) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12) + P(X = 13) + P(X = 14) + P(X = 15)
Using the binomial probability formula, calculate each individual probability and sum them up accordingly to find the probabilities for scenarios (b) to (e).