Respuesta :
Answer:
c. The area to the right of 4.5
Step-by-step explanation:
The normal approximation to the binomial is used to solve this question.
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, which is also there are under the normal curve to the left of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X, that is, the area under the normal curve to the right of 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].
We have that:
The binomial distribution is discrete.
The normal is continuous.
So, for the approximation, continuity correction is needed.
Then, with continuity correction, what we have is [tex]P(X > 4 + 0.5) = P(X > 4.5)[/tex], which is the area to the right of 4.5. The correct answer is c.
