Final answer:
The probability that at least 273 out of 350 voters will vote for a candidate using normal distribution approximation requires calculating the mean and standard deviation of the binomial distribution, converting to a z-score, and using standard normal distribution tables or a calculator to find the corresponding probability.
Step-by-step explanation:
To use the normal distribution to approximate the binomial distribution for the probability that at least 273 out of 350 voters will vote for a candidate, given that 73% of the voters are expected to vote for them, we must first determine the mean (μ) and standard deviation (σ) of the binomial distribution.
The formulas for these are μ = np and σ = √npq, where n is the number of trials, p is the probability of success, and q is the probability of failure (1 - p).
In this case, n = 350 and p = 0.73, so q = 0.27. Therefore, the mean is μ = 350 * 0.73 = 255.5, and the standard deviation is σ = √(350 * 0.73 * 0.27) ≈ 8.6745.
To find the probability of getting at least 273 voters, we convert this to a z-score in a normal distribution:
Z = (X - μ) / σ
Where X is the value we’re interested in. So, Z = (273 - 255.5) / 8.6745 ≈ 2.0173.
Using standard normal distribution tables or calculator, we find the probability to the left of this z-score and subtract it from 1 to get the probability of at least 273 voters.
Therefore, we find the area to the right of the z-score 2.0173, which corresponds to the probability that at least 273 voters will vote for the candidate.