Question

n a population of 500 voters, 40% belong to Party X. A simple random sample of 60 voters is taken. What is the chance that a majority (more than 50%) of the sampled voters belong to Party X

Answer 1

The normal approximation to the binomial distribution can be used to solve this:
The mean = np =60 * 0.4 = 24
The standard deviation = square root of [np(1 - p)] =
\[\sqrt{60\times 0.4\times 0.6}=3.795\]
The z-score for 30 voters is given by
\[z=\frac{30-24}{3.795}=1.581\]
From a standard normal distribution table it is found that the cumulative probability of up to 30 voters in the sample belonging to party X is 0.943. Therefore the probability that that a majority (more than 50%) of the sampled voters belong to Party X is given by 1 - 0.943 = 0.057 or 5.7%

Answer 2

Please find below a recalculation based on using the term 0.5 in the calculation of the z-score. The term 0.5 is a correction caused by the change from a discrete to a continuous distribution.
The z-score for 30 voters is given by
\[z=\frac{30-24+0.5}{3.795}=1.7127\]
From a standard normal distribution table it is found that the cumulative probability of up to 30 voters in the sample belonging to party X is 0.9567. Therefore the probability that that a majority (more than 50%) of the sampled voters belong to Party X is given by 1 - 0.9567 = 0.0433 or 4.33%

Answer 3

All the answers above are wrong.

N=500, G=200, n=60, g<=31
You have 500 voters and 40% or 200 of them belongs to party X. You sample 60 voters and want to have more than 30 or 31 of them to belong to the 200 voters for party X.
The probability for the event above to occur is: ((G,g)(N-G,n-g)/(N,n))

Answer 4

Using the hypergeometric distribution to calculate the probability that a majority (more than 50%) of the sampled voters belong to Party X gives a result approximately equal to 0.035 or 3.5%