Mathematics, 12.06.2021 23:30 kwhitt

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

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Answer from: himatasy

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

Step-by-step 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:

$Z = \frac{X - \mu}{\sigma}$

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 $s = \frac{\sigma}{\sqrt{n}}$ .

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 $s = \sqrt{\frac{p(1-p)}{n}}$

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 $p = 0.54, s = \sqrt{\frac{0.54*0.46}{450}} = 0.0235$

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

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{0.483 - 0.54}{0.0235}$

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.

Answer from: Quest

gimmee one of those gold crowns!

~coco

Answer from: Quest

too hard the other guy did it

step-by-step explanation:

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