A candidate for public office has claimed that 60% of voters will vote for her. If 5 registered voters were sampled,
1. What is the probability distribution function for this voting?
2. What is the probability that exactly 3 would say they favor this candidate?
3. Calculate the mean and standard deviation of the probability distribution
To answer these questions, we need to assume that the candidate's claim is true and that each voter's decision is independent of the others. We can use the binomial distribution to model this scenario.
1. The probability distribution function (PDF) for the number of voters who favor the candidate can be calculated using the binomial distribution formula:
P(X = k) = (n choose k) * p^k * (1-p)^(n-k)
Where:
- P(X = k) is the probability that exactly k voters favor the candidate,
- n is the sample size (in this case 5 since we are sampling 5 voters),
- k is the number of voters favoring the candidate,
- p is the probability that a voter favors the candidate (0.60 in this case),
- (n choose k) is the binomial coefficient, calculated as n! / (k! * (n-k)!)
2. To find the probability that exactly 3 out of 5 voters favor the candidate, we can substitute n = 5 and k = 3 into the PDF formula and calculate:
P(X = 3) = (5 choose 3) * (0.60)^3 * (1-0.60)^(5-3)
The binomial coefficient (5 choose 3) can be calculated as 5! / (3! * (5-3)!), which simplifies to 10.
P(X = 3) = 10 * 0.60^3 * 0.40^2
3. The mean and standard deviation of a binomial distribution can be calculated using the following formulas:
Mean (μ) = n * p
Standard Deviation (σ) = sqrt(n * p * (1-p))
In this case, n = 5 and p = 0.60:
Mean (μ) = 5 * 0.60
Standard Deviation (σ) = sqrt(5 * 0.60 * (1-0.60))
Now, we can calculate the values for mean and standard deviation.