The switchboard in a Minneapolis law office gets an average...

The switchboard in a Minneapolis law office gets an average of 5.5 incoming phone calls during the noon hour on Mondays. Experience shows that the existing staff can handle up to six calls in an hour. Let $X=$ the number of calls received at noon. a. Find the mean and standard deviation of $X$. b. What is the probability that the office receives at most six calls at noon on Monday? c. Find the probability that the law office receives six calls at noon. What does this mean to the law office staff who get, on average, 5.5 incoming phone calls at noon? d. What is the probability that the office receives more than eight calls at noon?

Transcript

00:01 This question gives us the information that a particular law office receives an average of 5 .5 calls at noon, where x is then measuring the number of calls received at noon.

00:12 So then x follows a poisson distribution with a mean number of calls at 5 .5.

00:21 Part a of the question asks us to find the mean and the standard deviation of this variable.

00:26 So as previously stated, the mean is 5 .5 calls.

00:30 And to find the standard deviation for a poisson distribution, it is the square root of the mean, so the square root of 5 .5, which works out to be 2 .35.

00:44 Part b on the question asks us to find what is the probability that the office receives at most six calls at noon on monday.

00:55 So at most six calls would mean that six is the greatest number, but we could have anything less than that.

01:03 So at most six would translate to finding the probability that x is less than or equal to six.

01:10 To calculate this, we can use a poisson distribution in our ti83 calculators or a similar statistical package.

01:19 In the calculators, we would use the poisson cdf.

01:23 And what we need to enter into the cdf is the mean of the distribution, which is 5 .5.

01:30 And then in the cdf, when it asks for an x value, the second number we enter in, that x value is the largest number that we want included in our probability answer.

01:43 So the largest number we want included is six.

01:46 So we'll put in for our x value six.

01:49 As we then execute that command, we find the probability of being .686...

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