Question

A poll reported 42% of voters favor the Republican candidate in an upcoming election. Assume that this percentage is true for the current population of all registered voters. Calculate the probability that less than 45% of a sample of 40 voters will vote for the Republican candidate. (Hint: you need to consider this as a sampling distribution).

Answer 1

Answer:

64.80% probability that less than 45% of a sample of 40 voters will vote for the Republican candidate.

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, the zscore 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 pvalue, we get the probability that the value of the measure is greater than X.

Considering a proportion p in a sample of size n as a sampling distribution, we have that $\mu = p, \sigma = \sqrt{\frac{p(1-p)}{n}}$

In this problem, we have that:

$p = 0.42, n = 40$

So

$\mu = 0.42, \sigma = \sqrt{\frac{0.42*0.58}{40}} = 0.0780$

Calculate the probability that less than 45% of a sample of 40 voters will vote for the Republican candidate.

This is the pvalue of Z when X = 0.45. So

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

$Z = \frac{0.45 - 0.42}{0.0780}$

$Z = 0.38$

$Z = 0.38$ has a pvalue of 0.6480

64.80% probability that less than 45% of a sample of 40 voters will vote for the Republican candidate.