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

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

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 = (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{(p(1-p))/(n)}`

In this problem, we have that:

`p = 0.42, n = 40`

So

`\mu = 0.42, \sigma = \sqrt{(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 = (X - \mu)/(\sigma)`

`Z = (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.