If 30% of people in an area are registered voters and 15 people are selected at random from the area:
A. what is the probability that at most 3 out of the 15 people are registered voters?
B. could we have used the normal approximation to the binomial in part b? Explain why or why not.
a)
P(0 of 15) =
P(1 of 15) =
P(2 of 15) =
P(3 of 15) =
then add these four probabilities together => P(at most 3 of 15);
b. I believe that we cannot because the sample of 15 people is too small (the critical size ia about 30 items).
Thank you so much Kuai. This helped me out
To calculate the probability, we need to use the binomial probability formula because we are dealing with a situation where there are only two possible outcomes (registered voter or not registered voter) and the probability of success (being a registered voter) is fixed at 30%.
A. To find the probability that at most 3 out of the 15 people are registered voters, we need to calculate the cumulative probability of having 0, 1, 2, or 3 registered voters. This can be done using the binomial probability formula:
P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
where X represents the number of registered voters.
Using the formula, we can calculate each individual probability:
P(X = 0) = C(15, 0) * 0.3^0 * (1 - 0.3)^(15-0)
P(X = 1) = C(15, 1) * 0.3^1 * (1 - 0.3)^(15-1)
P(X = 2) = C(15, 2) * 0.3^2 * (1 - 0.3)^(15-2)
P(X = 3) = C(15, 3) * 0.3^3 * (1 - 0.3)^(15-3)
where C(n, r) represents the combination function (nCr) which calculates the number of ways to choose r objects from a set of n objects.
B. In this case, we cannot use the normal approximation to the binomial distribution. The normal approximation assumes a sufficiently large sample size and a probability of success that is not close to 0 or 1. In our case, we have a relatively small sample size (15) and a probability of success (0.3) that is not close to 0 or 1. Therefore, a normal approximation would not be suitable. We should stick to using the binomial distribution to calculate probabilities in this scenario.