Question

The proportion of eligible voters in the next election who will vote for the incumbent is assumed to be 54%. What is the probability that in a random sample of 450 voters, less than 48.3% say they will vote for the incumbent

Answer 1

0.0076 = 0.76% probability that less than 48.3% say they will vote for the incumbent.

Explanation:

To solve this question, we use the normal probability distribution and the central limit theorem.

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the p-value, we get the probability that the value of the measure is greater than X.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean
`\mu` and standard deviation
`\sigma`, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
`\mu` and standard deviation
`s = (\sigma)/(√(n))`.

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean
`\mu = p` and standard deviation
`s = \sqrt{(p(1-p))/(n)}`

The proportion of eligible voters in the next election who will vote for the incumbent is assumed to be 54%. Sample of 450 voters.

This means that
`p = 0.54, s = \sqrt{(0.54*0.46)/(450)} = 0.0235`

What is the probability that in a random sample of 450 voters, less than 48.3% say they will vote for the incumbent?

This is the p-value of Z when X = 0.483. So

`Z = (X - \mu)/(\sigma)`

By the Central Limit Theorem

`Z = (X - \mu)/(s)`

`Z = (0.483 - 0.54)/(0.0235)`

`Z = -2.43`

`Z = -2.43` has a p-value of 0.0076

0.0076 = 0.76% probability that less than 48.3% say they will vote for the incumbent.