Respuesta :
Answer:
0.0409 = 4.09% probability that the company will receive more than 20 calls per hour
Step-by-step explanation:
To solve this question, we need to understand the Poisson distribution and the normal distribution.
Poisson distribution:
random variable is given by the following formula:
[tex]P(X = x) = \frac{e^{-\lambda}*\lambda^{x}}{(x)!}[/tex]
In which
x is the number of sucesses
e = 2.71828 is the Euler number
[tex]\lambda[/tex] is the mean in the given interval, which is the same as the variance.
Normal distribution:
When the distribution is normal, we use 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.
The normal approximation can be used to a Poisson distribution, with [tex]\mu = \lambda, \sigma = \sqrt{\lambda}[/tex]
Computer Help Hot Line receives, on average, 14 calls per hour asking for assistance.
This means that [tex]\lambda = 14[/tex].
Then
[tex]\mu = 14, \sigma = \sqrt{14} = 3.74[/tex]
What is the probability that the company will receive more than 20 calls per hour?
Using continuity correction, this is [tex]P(X > 20 + 0.5) = P(X > 20.5)[/tex], which is 1 subtracted by the pvalue of Z when X = 20.5. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{20.5 - 14}{3.74}[/tex]
[tex]Z = 1.74[/tex]
[tex]Z = 1.74[/tex] has a pvalue of 0.9591
1 - 0.9591 = 0.0409
0.0409 = 4.09% probability that the company will receive more than 20 calls per hour
