Exit polling is a popular technique used to determine the outcome of an election prior to results being tallied. Suppose a referendum to increase funding for education is on a ballot in a large town (voting population over 100,000). An exit poll of 300 voters finds that 144 voted for the referendum.
How likely are the results of your sample if the population proportion of voters in the town in favor of the referendums is 0.52?
The probability that less than 144 people voted for the referendum is___________ (round to four decimal places)
To determine the likelihood of the results of the sample, we need to use the binomial probability distribution.
The binomial distribution is used when we have a fixed number of independent trials (in this case, 300 voters) and each trial has two possible outcomes (voting for or against the referendum). The probability of success for each trial (p) is given as 0.52.
The probability of getting exactly k successes (in this case, 144) out of n trials can be calculated using the formula for binomial probability:
P(X = k) = nCk * p^k * (1-p)^(n-k)
Where nCk represents the combination formula for selecting k successes out of n trials.
Therefore, to calculate the probability of getting less than 144 people voting for the referendum, we need to sum up the probabilities for k=0 to k=143.
P(X < 144) = P(X = 0) + P(X = 1) + ... + P(X = 143).
Calculating this sum would require a lot of calculations. Fortunately, there are statistical tools and software available that can calculate this probability for us. We can use Python, R, or online calculators to find the cumulative probability.
Using an online binomial probability calculator or a statistical tool, we can find the cumulative probability P(X < 144) given the parameters n = 300, p = 0.52, and k = 143.
Using the cumulative probability, we can find that the probability that less than 144 people voted for the referendum is approximately 0.3032 (rounded to four decimal places).
Note: It is important to keep in mind that exit polls provide an estimate and have some level of uncertainty. The probability calculation evaluates the likelihood of obtaining results as extreme as the observed data if the population proportion of voters in favor of the referendum is 0.52.
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Thanks I have the answer now
Population has binomial distribution.
sample mean is 144/300.
Check that sample satisfies normal approximation and proceed as in your next post.
Do not forget to apply continuity correction when approximating a discrete distribution (such as binomial) by a continuous distribution (such as normal).