In the last senatorial election in New Jersey, the republican candidate got 48% of the vote. During the election an exit poll was conducted by radio station WSTAT. Ten voters selected at random where asked if they voted for the republican candidate.
(a) Let X = "Number of votes for the republican candidate". Give the distribution of X and explain why.
(b) Compute the P(X = 5), P(X < 6), and P(X > 5). Do the calculation directly using doing the calculation directly from the formula or using the Binomial table enclosed.
(c) Give the mean and variance of X. Can we use the normal approximation to the binomial?
(d) Use the normal approximation (regardless of your answer in (c)) to compute the probabilities the P(X = 5), P(X < 6), and P(X > 5) and compare the results with those in part (b).
(a) The distribution of X, the number of votes for the republican candidate, follows a binomial distribution. This is because there are only two possible outcomes for each voter -- either they vote for the republican candidate or they don't. Additionally, each voter's choice is independent of others and the probability of voting for the republican candidate remains constant at 48% for each voter.
(b) To compute the probabilities P(X = 5), P(X < 6), and P(X > 5) directly using the binomial distribution formula, we need to know the total number of voters in the sample and the probability of success (voting for the republican candidate).
For example, if there were a total of 10 voters in the sample and the probability of voting for the republican candidate is 0.48, we can calculate the probabilities as follows:
P(X = 5) = (10 choose 5) * (0.48^5) * ((1-0.48)^(10-5))
P(X < 6) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4) + P(X = 5)
P(X > 5) = 1 - P(X < 6)
You can refer to a binomial table to find the values of combinations (10 choose 5) and calculate the probabilities using the formula.
(c) The mean of X can be calculated using the formula μ = np, where n is the number of trials (total voters) and p is the probability of success (voting for the republican candidate). In this case, μ = 10 * 0.48 = 4.8.
The variance of X can be calculated using the formula σ^2 = np(1-p). Therefore, σ^2 = 10 * 0.48 * (1-0.48) = 2.496.
To determine if we can use the normal approximation to the binomial, we can use the guidelines commonly used in practice. If np and n(1-p) are both greater than or equal to 5, the normal approximation can be considered reasonably accurate. In this case, since np = 4.8 and n(1-p) = 5.2, we can use the normal approximation.
(d) To compute the probabilities using the normal approximation, we can use the mean and variance obtained in part (c) and apply the z-score formula.
For P(X = 5):
- Calculate the z-score using Z = (X - μ) / σ, where X is the value of interest, μ is the mean, and σ is the standard deviation.
- Find the corresponding probability from the standard normal distribution table using the z-score.
For P(X < 6) and P(X > 5):
- Calculate the z-scores for 6 using the same method as above.
- Use the z-score values to find the corresponding probabilities from the standard normal distribution table.
Comparing the results from part (b) (using the binomial table) and part (d) (using the normal approximation) will give us an idea of the accuracy of the approximation.