Respuesta :
Answer:
C. 0.0207
Step-by-step explanation:
For each person, there are only two possible outcomes. Either they vote for Carl, or they vote for Nathan. So we use the binomial probability distribution.
However, we are working with samples that are considerably big. So i am going to aproximate this binomial distribution to the normal.
Binomial probability distribution
Probability of exactly x sucesses on n repeated trials, with p probability.
Can be approximated to a normal distribution, using the expected value and the standard deviation.
The expected value of the binomial distribution is:
[tex]E(X) = np[/tex]
The standard deviation of the binomial distribution is:
[tex]\sqrt{V(X)} = \sqrt{np(1-p)}[/tex]
Normal probability distribution
Problems of normally distributed samples can be 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.
When we are approximating a binomial distribution to a normal one, we have that [tex]\mu = E(X)[/tex], [tex]\sigma = \sqrt{V(X)}[/tex].
In this problem, we have that:
[tex]\mu = 100*0.4 = 40[/tex]
[tex]\sigma = \sqrt{100*0.4*0.6} = 4.9[/tex]
What is the likelihood that the poll will show a majority in favor of Carl?
This is 1 subtracted by the pvalue of Z when X = 50. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{50 - 40}{4.9}[/tex]
[tex]Z = 2.04[/tex]
[tex]Z = 2.04[/tex] has a pvalue of 0.9793. So the answer is 1-0.9793 = 0.0207.
