Question 1169101: You start a new job answering phones, and when you’re hired, your boss tells you
that you should expect about 4 calls each hour. You notice that you are regularly
getting more than 8 calls each hour. If the average number of calls per hour
is 4, what is the probability that you get more than 8 calls in an hour? Do you
think your boss is being honest with you?
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
mu = average number of calls per hour = 4
We'll use the Poisson Distribution
The formula is
P(x) = (mu^x*e^(-mu))/(x!)
where the exclamation mark indicates factorial. The variable x is any positive whole number.
So if we wanted to find the probability of getting exactly x = 0 calls per hour, then we would say
P(x) = (mu^x*e^(-mu))/(x!)
P(0) = (4^0*e^(-4))/(0!)
P(0) = 0.0183156388887
If we wanted the probability of getting 1 call per hour, then,
P(x) = (mu^x*e^(-mu))/(x!)
P(1) = (4^1*e^(-4))/(1!)
P(1) = 0.07326255555483
We continue this for x = 2, x = 3, all the way up to x = 8
Once we have all of those P(x) values, we add up the results to get P(X <= 8) which represents the probability of getting at most 8 calls per hour.
Doing this is a bit tediuous, but you can use the poissonCDF function in your TI83 or TI84 calculator to speed up the process. You can also use an online calculator like this
https://stattrek.com/online-calculator/poisson.aspx
to get the job done
Whichever method you pick you should find that P(X <= 8) is roughly 0.97864, so 1-0.97864 = 0.02136 is the approximate probability of getting more than 8 calls in an hour.
There's roughly a 2.136% chance of getting more than 8 calls per hour, assuming the claim "the mean number of calls per hour is 4" is correct.
Because this event of getting more than 8 calls per hour happens regularly, I would say that your boss is not being honest; or perhaps they are just using the wrong information. Either way, the average number of calls appears to be greater than 4.
Side note: if you're doing a hypothesis test with significance level alpha = 0.05, then we would reject the null hypothesis and conclude that the mean is larger than 4. Recall that you always reject the null if the p value is smaller than alpha. The p value is equal to the probability of getting some event, plus any extreme beyond that event. The p value in this case is roughly 0.02136
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