Question

Computer Help Hot Line receives, on average, 14 calls per hour asking for assistance. Assume the variable follows a Poisson distribution. What is the probability that the company will receive more than 20 calls per hour? Round answer to 4 decimal places. Answer:

Answer 1

0.0409 = 4.09% probability that the company will receive more than 20 calls per hour

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:

`P(X = x) = (e^(-\lambda)*\lambda^(x))/((x)!)`

In which

x is the number of sucesses

e = 2.71828 is the Euler number

`\lambda` 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
`\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.

The normal approximation can be used to a Poisson distribution, with
`\mu = \lambda, \sigma = √(\lambda)`

Computer Help Hot Line receives, on average, 14 calls per hour asking for assistance.

This means that
`\lambda = 14`.

Then

`\mu = 14, \sigma = √(14) = 3.74`

What is the probability that the company will receive more than 20 calls per hour?

Using continuity correction, this is
`P(X > 20 + 0.5) = P(X > 20.5)`, which is 1 subtracted by the pvalue of Z when X = 20.5. So

`Z = (X - \mu)/(\sigma)`

`Z = (20.5 - 14)/(3.74)`

`Z = 1.74`

`Z = 1.74` 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