Respuesta :
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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 [tex]\mu = p, \sigma = \sqrt{\frac{p(1-p)}{n}}[/tex]
In this problem, we have that:
[tex]p = 0.42, n = 40[/tex]
So
[tex]\mu = 0.42, \sigma = \sqrt{\frac{0.42*0.58}{40}} = 0.0780[/tex]
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
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{0.45 - 0.42}{0.0780}[/tex]
[tex]Z = 0.38[/tex]
[tex]Z = 0.38[/tex] 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.
