SOLUTION: On the midnight shift, the number of patients with head trauma in an emergency room has the probability distribution shown below.
x 0 1 2 3 4 5 Total
P(x) .03 .42 .30 .19 .0
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Question 1196800: On the midnight shift, the number of patients with head trauma in an emergency room has the probability distribution shown below.
x 0 1 2 3 4 5 Total
P(x) .03 .42 .30 .19 .05 .01 1.00
(a) Calculate the mean and standard deviation. (Round your mean value to 2 decimal places and standard deviation to 3 decimal places.)
Answer by math_tutor2020(3817) (Show Source): You can put this solution on YOUR website!
Answers:
Mean = 1.84
Standard Deviation = 0.997
The mean is exact, while the standard deviation is approximate.
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Work Shown:
We'll use this table to compute the mean
| x | 0 | 1 | 2 | 3 | 4 | 5 | Total |
| P(x) | 0.03 | 0.42 | 0.3 | 0.19 | 0.05 | 0.01 | 1 |
| x*P(x) | 0 | 0.42 | 0.6 | 0.57 | 0.2 | 0.05 | 1.84 |
The x*P(x) row refers to multiplying each column of x and P(x) values.
Then we add up all of the x*P(x) values to get 1.84 which is the mean.
The Greek letter mu is often used as the mean
So we have mu = 1.84 and this value is useful to find the standard deviation.
| x | 0 | 1 | 2 | 3 | 4 | 5 | Total |
| P(x) | 0.03 | 0.42 | 0.3 | 0.19 | 0.05 | 0.01 | 1 |
| X-mu | -1.84 | -0.84 | 0.16 | 1.16 | 2.16 | 3.16 | |
| (x-mu)^2 | 3.3856 | 0.7056 | 0.0256 | 1.3456 | 4.6656 | 9.9856 | |
| (x-mu)^2*P(x) | 0.101568 | 0.296352 | 0.00768 | 0.255664 | 0.23328 | 0.099856 | 0.9944 |
The value 0.9944 in the bottom right corner represents the sum of the (x-mu)^2*P(x) values
This value also represents the variance of the probability distribution table.
Take the square root of the variance to get the standard deviation
standard deviation = sqrt(variance)
standard deviation = sqrt(0.9944)
standard deviation = 0.99719606898543
standard deviation = 0.997
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