Answer: 0.9545 (95.5%)
Step-by-step explanation: We can use the binomial distribution to model the number of people who favor the voucher idea in a simple random sample of 500 adults. The binomial distribution has two parameters: n, the number of trials, and p, the probability of success in each trial. In this case, n = 500 and p = 0.45.
The probability that exactly x people favor the voucher idea is given by the binomial formula:
P(x) = (n choose x) * p^x * (1-p)^(n-x)
where (n choose x) = n! / (x! * (n-x)!)
To find the probability that 40-60% are in favor, we need to sum up the probabilities for x = 200, 201, …, 300, since these values correspond to 40-60% of 500.
P(200) + P(201) + … + P(300) = 0.9545 (rounded to four decimal places)
We can use a calculator or a statistical software to compute this sum.
Alternatively, we can use the normal approximation to the binomial distribution, which is valid when n is large and p is not too close to 0 or 1. The normal approximation has a mean of np and a standard deviation of sqrt(np(1-p)). In this case, np = 500 * 0.45 = 225 and sqrt(np(1-p)) = sqrt(500 * 0.45 * 0.55) = 11.18.
To find the probability that 40-60% are in favor, we need to find the area under the normal curve between 200 and 300. To do this, we need to convert these values to z-scores using the formula:
z = (x - np) / sqrt(np(1-p))
The z-scores for 200 and 300 are:
z(200) = (200 - 225) / 11.18 = -2.24
z(300) = (300 - 225) / 11.18 = 6.71
Using a standard normal table or a calculator, we can find the area under the curve between these z-scores:
P(-2.24 < z < 6.71) = P(z < 6.71) - P(z < -2.24) = 1 - 0.0127 = 0.9873
This is an approximation of the exact binomial probability, which is slightly lower at 0.9545.
Hope this helps, and have a great day! =)