an opinion poll asks a sample of 500 adults (an SRS) whether they favor giving parents of school age children vouchers that can be exchanged for education at any public or private school of their choice. Each school would be paid by the government on the basis of how many vouchers it collected. suppose that in fact 45% of the population favor this idea. what is the probability that more than half of the sample are in favor
To find the probability that more than half of the sample are in favor, we can use the binomial probability formula. The binomial probability formula is used when there are two possible outcomes (favor or not favor) and the sample size is fixed (500 adults).
First, we need to calculate the probability of an individual favoring the idea. We are given that 45% of the population favors this idea, so the probability of an individual favoring is 0.45.
Next, we need to use this probability along with the sample size to calculate the probability of more than half of the sample favoring. We will use the cumulative binomial distribution formula to find this probability:
P(X > x) = 1 - P(X ≤ x)
Where:
P(X > x) is the probability of more than x individuals favoring,
P(X ≤ x) is the cumulative probability of x or fewer individuals favoring,
and x is the number of individuals (in this case, half of the sample size).
Given that the sample size is 500, we need to calculate P(X ≤ 250). This is the cumulative probability of 250 or fewer individuals favoring.
We can use a statistical software tool, such as R or Excel, to calculate this probability. Alternatively, we can use a binomial probability table or calculator.
Let's assume we are using a statistical software tool and it gives us the cumulative probability: P(X ≤ 250) = 0.510.
Therefore, the probability of more than half of the sample favoring is:
P(X > 250) = 1 - P(X ≤ 250) = 1 - 0.510 = 0.490.
So, the probability that more than half of the sample are in favor is 0.490 or 49%.